Revision as of 17:20, 9 May 2016

1 INTRODUCTION

Much effort has been spent in developing the so called stabilized numerical methods overcoming the two main sources of instability in incompressible flow analysis, namely those originated by the high values of the convective terms and those induced by the difficulty in satisfying the incompressibility condition.

The solution of above problems in the context of the finite element method (FEM) has been attempted in a number of ways. The first attempts to correct the underdiffusive character of the Galerkin FEM for high convection flows were based in adding some kind of artificial viscosity terms to the standard Galerkin equations 1.

A popular way to overcome the problems with the incompressibility constraint is by introducing a pseudo-compressibility in the flow and using implicit and explicit algorithms developed for this kind of problems such as artificial compressibility schemes 3 and preconditioning techniques 6. State of the art FEM schemes for fluid flow analysis with good stabilization properties for the convective and incompressibility terms are based in Petrov-Galerkin (PG) techniques. The background of PG methods are the non-centred (upwind) schemes for computing the first derivatives of the convective operator in finite difference and finite volume methods 7. A general class of stabilized PG FEM has been recently developed where the standard Galerkin variational form of the momentum and mass balance equations is extended with adequate residual-based terms in order to achieve a stabilized numerical scheme. References 9 list some of the more popular stabilized FEM of this kind. A review of many of these methods can be found in 1.

In this paper a stabilized FEM for incompressible flows is derived via a finite calculus (FIC) approach 32. The FIC method is based in invoking the balance of fluxes in a fluid domain of finite size. This introduces naturally additional terms in the classical differential equations of momentum and mass balance of infinitesimal fluid mechanics which are a function of characteristic length dimensions related to the element size in the discretized problem. The FIC terms in the modified governing equations provide the necessary stabilization to the discrete equations obtained via the standard Galerkin FEM. The FIC/FEM formulation allows to use low order finite elements (such as linear triangles and tetrahedra) with equal order approximations for the velocity and pressure variables.

The FIC/FEM formulation has proven to be very effective for the solution of a wide class of problems, such as convection-diffusion 32 and convection-diffusion-reaction 40 involving arbitrary high gradients, incompressible flow problems accounting for free surface effects and fluid-structure interaction situations 32 and quasi and fully incompressible problems in solid mechanics 52.

The FIC equations for incompressible flow derived in previous works of the authors assumed that the dimensions of the domain where the momentum conservations law was enforced remain the same independently of the direction along which balance of momentum is imposed. As a consequence, each of the resulting FIC momentum equations contain the same characteristic dimensions which can be grouped in a characteristic distance vector. In this paper, a refined FIC momentum equations are derived by accepting that the dimensions of the momentum balance domain are different for each of the momentum equations. This introduces a matrix form of the characteristic distances and of the corresponding FIC terms which have better intrinsic stabilization properties.

The idea of a matrix form of the stabilization parameters is close to the element-matrix-based and element-vector-based stabilization parameters proposed in 56 where different intrinsic time parameters wsere defined separately for each degree of freedom of the equation system.

Stabilized FEM have been successfully used in the past to solve a wide range of fluid mechanics problems. The intrinsice dissipative properties of the stabilization terms (which can interpreted as an additional viscosity) typically suffices to yield good results for low and moderate values of the Reynolds number (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle Re} ). For high values of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle Re}

most stabilized FEM fail to provide physically meaningful results and the numerical solution is often unstable or inaccurate. The introduction of a turbulence model  is mandatory in order to obtain meaningful results in these cases.

The relationship between the additional dissipation introduced by the turbulence model and the intrinsic dissipative properties of stabilized FEM is an open topic which is attracting increasing attention in the CFD community. It is clear that both remedies (the turbulence model and the stabilization terms) play a similar role in the numerical solution, i.e. that of ensuring a solution which is ``physically sound´´ and as accurate as possible.

It is our belief that the matrix stabilization terms introduced by the FIC/FEM formulation here presented allow to model accurately high Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle Re}

number flows without the need of introducing any turbulence model. The background of this belief originates in the positive experiences in the application of a very similar formulation for solving advection-diffusion and advection-diffusion-reaction problems with arbitrary sharp gradients without introducing any transverse dissipation terms 39. The extension of these ideas to the Navier-Stokes equations described here provides a straightforward procedure for solving a wide class of flow problems from low to high Reynolds numbers, as demonstrated by the good results presented in the paper.

The layout of the paper is the following. In the next section the FIC equations for incompressible flows with matrix stabilization terms are presented. The finite element discretization is introduced and the resulting matrix equations are detailed. A fractional step scheme for the transient solution is detailed. Examples of applications to the 2D analysis of flows passing a backward facing step and a cylinder at different Reynolds numbers are presented.

2 GENERAL FIC EQUATIONS FOR VISCOUS INCOMPRESSIBLE FLOW

The FIC governing equations for a viscous incompressible fluid can be written in an Eulerian frame of reference as

Momentum

Mass balance

where

Above Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \Omega }

is the analysis domain, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle n_d}
is the number of space dimensions (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle n_d=2}
for 2D problems), Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle u_i}
is the velocity along the ith global axis, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \rho }
is the (constant) density of the fluid, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle p}
is the absolute pressure (defined positive in compression), Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle b_i}
are the body forces and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle s_{ij}}
are the viscous deviatoric stresses related to the viscosity Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \mu }
by the standard expression

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \delta _{ij}}

is the Kronecker delta and the strain rates Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \dot \varepsilon _{ij}}
are

The FIC boundary conditions are

and the initial condition is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle u_j =u_j^0}

for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle t=t_0}

.

Summation convention for repeated indices in products and derivatives is used unless otherwise specified.

In Eqs.(13) and (14) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle t_i}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle u_j^p}
are surface tractions and prescribed displacements on the boundaries Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \Gamma _t}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \Gamma _u}

, respectively, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle n_j}

are the components of the unit normal vector to the boundary and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \sigma _{ij}}
are the total stresses given by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \sigma _{ij}=s_{ij}-\delta _{ij}p}

.

The Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle h_{ij}}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle h_j}
are characteristic distances of the domain where balance of momentum and mass is enforced. In Eq.(7) these lengths define the domain where equilibrium of boundary tractions is established 32. In the discretized problem the characteristic distances become of the order of the typical element dimensions. Note that by making these distances equal to zero the standard infinitessimal form of the fluid mechanics equations is recovered 1.

Eqs.(1)–(6) are the starting point for deriving stabilized FEM for solving the incompressible Navier-Stokes equations. The underlined FIC terms in Eq.(1) are essential to overcome the numerical instabilities due to the convective terms in the momentum equations, whereas the underlined terms in Eq.(2) take care of the instabilities due to the incompressibility constraint. An interesting feature of the FIC formulation is that it allows to use equal order interpolation for the velocity and pressure variables 43.

Remark 1. In previous work of the authors the characteristic distances in the momentum equations had a vector form, i.e. the FIC momentum equations were written as

or

where (for 2D problems) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {h}=[h_1,h_2]^T}

is the characteristic length vector 32.

The difference of Eqs.(7) with Eq.(1) is that the characteristic distances have now a matrix form, i.e. the expanded form of the momentum equations (1) is (for 2D problems)

The rationale of Eqs.(8) is briefly explained in the Appendix.

The matrix of stabilization parameters H is defined as (for 2D problems)

Remark 2. Note that the characteristic distances in the FIC mass conservation equation (2) have a vector form. As mentioned above distances Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle h_1}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle h_2}
in Eq.(2) (for 2D problems) denote the dimensions of the domain where balance of mass is globally enforced (see Appendix). This is a basic difference with the momentum equations where the momentum balance law is applied along each global coordinate direction.

2.1 Stabilized integral forms

From the momentum equations it can be obtained 43

where

Substituting Eq.(15) into Eq.(8) and retaining the terms involving the derivatives of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle r_{m_i}}

with respect to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x_i}
only, leads to the following alternative expression for the stabilized mass balance equation

with

The Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \tau _i} 's in Eq.(11) when multiplied by the density are equivalent to the intrinsic time parameters, seen extensively in the stabilization literature. The interest of Eq.(11) is that it introduces the first space derivatives of the momentum equations into the mass balance equation. These terms have intrinsic good stability properties as explained next.

The weighted residual form of the momentum and mass balance equations (Eqs.(1) and (11)) is written as

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \delta u_i}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle q}
are arbitrary weighting functions representing virtual velocities and virtual pressure fields. Integrating by parts the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle r_{m_i}}
terms  in Eqs.(13) and (14) leads to

We will neglect hereonwards the third integral in Eq.(15b) by assuming that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle r_{m_i}}

is negligible on the boundaries. The deviatoric stresses and the pressure terms in the first integral of Eq.(15a) are integrated by parts in the usual manner. The resulting momentum and mass balance equations are

In the derivation of the viscous term in Eq.(16a) we have used the following identity holding for incompressible fluids (prior to the integration by parts)

2.2 Convective and pressure gradient projections

The computation of the residual terms are simplified if we introduce the convective and pressure gradient projections Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle c_i}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \pi _i}

, respectively defined as

We can express Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle r_{m_i}}

in  Eqs.(16a) and (16b) in terms of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle c_i}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \pi _i}

, respectively which then become additional variables. The system of integral equations is now augmented in the necessary number of equations by imposing that the residual Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle r_{m_i}}

vanishes (in average sense) for both forms given by Eqs.(18). This gives the final system of governing equation as:

with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle i,j,k=1,n_d} . In Eqs.(21) and (22) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \delta c_i}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \delta \pi _i}
are appropriate weighting functions and the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \rho }
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \tau _i}
weights are introduced for convenience.

We note that accounting for the convective and pressure gradient projections enforces the consistency of the formulation as it ensures that the stabilization terms in Eqs.(19) and (20) have a residual form which vanishes for the ``exact´´ solution. Neglecting these terms can reduce the accuracy of the numerical solution and it makes the formulation more sensitive to the value of the stabilization parameters as shown in references 51.

3 FINITE ELEMENT DISCRETIZATION

We choose Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle C^\circ }

continuous linear interpolations of the velocities, the pressure, the convection projections Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle c_i}
and the pressure gradient projections Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \pi _i}
over 3-noded triangles (2D) and 4-noded tetrahedra (3D). The linear interpolations are written as

where the sum goes over the number of nodes of each element Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle n}

(Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle n=3/4}
for triangles/tetrahedra), Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \bar {(\cdot )}^k}
denotes the nodal variables and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle N^k}
are the linear shape functions 1.

Substituting the approximations (23) into Eqs.(19)–(22) and choosing the Galerking form with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \delta u_i =q=\delta c_i=\delta \pi _i =N^i}

leads to following system of discretized equations

where

If we denote the node indexes with superscripts Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle a,b} , the space indices with subscripts Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle i,j} , the element contributions to the components of the arrays involved in these equations are (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle i,j=1,3}

for 3D problems)

It is understood that all the arrays are matrices (except Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {f}}

which is a vector) whose components are obtained by grouping together the left indices in the previous expressions (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle a}
and possibly Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle i}

) and the right indices (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle b}

and possibly Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle j}

).

Note that the stabilization matrix Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \hat{K}}

in Eq.(25) adds additional orthotropic diffusivity terms of value Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \rho \displaystyle{h_{ij}  u_l\over 2}}

.

The overall stabilization terms introduced by the FIC formulation above presented have the intrinsic capacity to ensure physically sound numerical solutions for a wide spectrum of Reynolds numbers without the need of introducing additional turbulence modelling terms. This interesting property is validated in the solution of the examples presented in a next section.

3.1 Transient solution scheme

The solution in time of the system of Eqs.(24) can be written in general form as

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {H}^{n+\theta }={H} ({u}^{n+\theta })} , etc and the parameter Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \theta \in [0,1]} . The direct monolitic solution of Eqs.(27) is possible using an adequate iterative scheme 52. However, in our work we have used the fractional step method described next.

4 FRACTIONAL STEP METHOD

A fractional step scheme is derived by noting that the discretized momentum equation (27a) can be split into the two following equations

In Eqs.(28) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \tilde{u}^{n+1}}

is a predicted value of the velocity at time Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle n+1}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \alpha }
is a variable whose values of interest are zero and one. For Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \alpha =0}
(first order scheme) the splitting error is of order Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 0 (\Delta t)}

, whereas for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \alpha =1}

(second order scheme) the error is of order Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 0 (\Delta t^2)}
52. We have chosen Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \alpha =1}
for the solution of the examples presented in the paper.

Eqs.(28) are completed with the following three equations emanating from Eqs.(27b-d)

The value of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \bar{u}^{n+1}}

obtained from Eq.(28b) is substituted into Eq.(29a) to give

The product Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {G}^T {M}^{-1} {G}}

can be approximated by a laplacian matrix, i.e.

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle L^{ab}}

are the element contributions to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {L}}

.

The steps of the fractional step scheme are:

Step 1

Eq.(28a) is linearized as

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \tilde{u}^{n+\theta }=\theta \tilde{u}^{n+1} + (1-\theta )\bar{u}^{n}} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \tilde{H}^{n+\theta } = {H} (\tilde{u}^{n+\theta })} , and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \tilde{C}^{n+\theta }= {C} (\tilde{u}^{n+\theta })} . We have chosen in our computation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \theta =0} . For this value, the fractional nodal velocities Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \tilde {u}^{n+1}}

can be  explicitely computed from Eq.(32) by

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {M}_d}

is the lumped diagonal form of M.

Step 2 Compute Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \bar{p}^{n+1}}

from Eq.(30) as

Step 3 Compute Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \bar{u}^{n+1}}

explicitly from Eq.(28a) as

Step 4 Compute Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \bar{c}^{n+1}}

explicitly from Eq.(29b) as

Step 5 Compute Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \bar{ \pi \hbox{ }}^{n+1}}

explicitly from Eq.(29c) as

Above algorithm has improved stabilization properties versus the standard segregation methods due to the introduction of the laplacian matrix Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \hat{L}}

in Eq.(34) which emanates from the FIC stabilization terms.

The boundary conditions are applied as follows. No condition is applied in the computation of the fractional velocities Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \tilde{u}^{n+1}}

in Eq.(33). The prescribed velocities at the boundary are applied when solving for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \bar{u}^{n+1}}
in the step 3. The prescribed pressures at the boundary are imposed by making Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \bar{p}^n}
equal to the prescribed pressure values.

5 STOKES FLOW

The formulation for a Stokes flow can be readily obtained simply by neglecting the convective terms in the general Navier-Stokes formulation. Consequently, the convective stabilization terms and the convective projection variables are not larger necessary. Also the intrinsic time parameters Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \tau _i}

take now the simpler form (see Eq.(12)):

The resulting discretized system of equations can be written as (see Eqs.(29))

The fractional step algorithm of the previous section can now be implemented. We note that convergence of the predictor-corrector scheme is now faster due to the absence of the non linear convective terms in the momentum equation.

The steady-state form of Eqs.(39) can be expressed in matrix form as

The system is symmetric and always positive definite and therefore leads to a non singular solution. This property holds for any interpolation function chosen for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \bar {u},\bar {p}}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \bar {              \pi \hbox{ }}}

, therefore overcoming the Babuŝka-Brezzi (BB) restrictions 1.

A reduced velocity-pressure formulation can be obtained by eliminating the pressure gradient projection variables Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \bar { \pi \hbox{ }}}

from the last equation  to give

The reduction process is simplified by using a diagonal form of matrix Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \hat {M}} . Applications of this scheme to incompressible solid mechanics problems are reported in 52.

6 COMPUTATION OF THE CHARACTERISTIC DISTANCES

The computation of the stabilization parameters is a crucial issue as they affect both the stability and accuracy of the numerical solution. The different procedures to compute the stabilization parameters are typically based on the study of simplified forms of the stabilized equations. Contributions to this topic are reported in 11. Despite the relevance of the problem there still lacks a general method to compute the stabilization parameters for all the range of flow situations.

Recent work of the authors in the application of the FIC/FEM formulation to convection-diffusion problems with sharp arbitrary gradients 39 has shown that the stabilizing FIC terms take the form of a simple orthotropic diffusion if the balance equation is written in the principal curvature directions of the solution. Excellent results were reported in 39 by computing first the characteristic length distances along the principal curvature directions, followed by a standard transformation of the distances to global axes. The resulting stabilized finite element equations capture the high gradient zones in the vicinity of the domain edges (boundary layers) as well as the sharp gradients appearing randomly in the interior of the domain 39. The FIC/FEM thus reproduces the best features of both the so called transverse (cross-wind) dissipation or shock capturing methods 58.

The numerical computations are simplified without apparent loss of accuracy if the main principal curvature direction of the solution at each element point is approximated by the direction of the gradient vector at the element center. The second principal direction (for 2D problems) is taken in the orthogonal direction to the gradient. For linear triangles and quadrilaterals these directions are assumed to be constant within the element 39.

Above simple scheme has been extended in this work for the computation of the characteristic distances Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle h_{ij}}

for the momentum equations. As for the length parameters Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle h_i}
in the mass conservation equation, the simplest assumption Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle h_i = h_{ii}}
has been taken. Details of the algorithm for computing Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle h_{ij}}
are given next (the method is explained for 2D problems although it is readily extendible to 3D problems).

() Definition of the principal curvature direction Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \vec \xi _1^i along the gradient of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): u_i .

Definition of the element characteristic distances Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): l_{i1} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): l_{i2} corresponding to the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): i th momentum equation.

For the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle i} -th momentum balance equation and every step of the fractional step method described in Section 4:

1. A coordinate system Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \vec \xi _1^i} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \vec \xi _2^i} is defined at each element point such that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \vec \xi ^1_i} is aligned with the gradient of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle u_i} (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \vec \xi _1^i = \vec{\nabla u_i}} ) and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \vec \xi _2^i} is orthogonal to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \vec \xi _1^i} in anticlockwise sense (Figure 1). The angle that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \vec \xi _1^i} forms with the global Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x_1} axis is defined as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \alpha _i} . Recall that upper and lower index Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle i} denotes the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle i} th momentum equation.
2. The element characteristic distances Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle l_{i1}} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle l_{i2}} are defined as the maximum projections of the element sides along the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \vec \xi _1^i} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \vec \xi _2^i} axes, respectively (Figure 2).
3. The characteristic distances Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle h_{i1}} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle h_{i2}} are computed as

   Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \left\{\begin{matrix}h_{i1}\\ h_{i2}\\\end{matrix}\right\}=\left[\begin{matrix}c_i & -s_i \\ s_i &c_i\\\end{matrix} \right]\left\{\begin{matrix}\bar h_{i1}\\ \bar h_{i2}\\\end{matrix}\right\}\quad , \quad i=1,2

   (41)

with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle c_i = \cos \alpha _i, s_i =\sin \alpha _i}

and the local distances Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \bar h_{i1}}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \bar h_{i2}}
are

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \bar h_{i1} =\left(\coth \bar \gamma _{ij} -{1\over \bar \gamma _{ij}} \right)l_{ij}\quad , \quad \gamma _{ij} = {\bar u_j l_{ij}\over 2\mu } \quad j=1,2

(42)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \bar u_1}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \bar u_2}
are the components of the velocity vector along the local axes Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \vec \xi _1^i}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \vec \xi _2^i}

, respectively (Figure 1).

7 EXAMPLES

The examples were solved with the Tdyn code where the formulation here presented has been implemented. The Tdyn code can be downloaded from the webpage given in [58].

7.1 Backwards facing step at high Reynolds number

Figure 3 shows the geometry of the standard backwards facing step problem. The boundary conditions were the following: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle u_1=1}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle u_2=0}
 were taken  at the entry while Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle p=0}
was assumed at the exit. Slipping conditions were assumed at the rest of the vertical and horizontal walls. A value of the kinematic viscosity Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \nu =\displaystyle{\mu \over \rho } =2.1 \times 10^{-5}}
was taken giving a Reynolds number of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle Re =\displaystyle{u_{free} H\over \nu } = 47619}
for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle H=1}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle u_{free}=1}

.

Figure 3 also shows the relatively coarse mesh chosen of 30850 three-noded triangular elements and 15426 nodes. The contours of the horizontal and vertical velocities and details of the velocity vectors are shown in Figures 4 and 5, respectively. Figure 6 shows the distribution of the horizontal velocity along the bottom line starting from the vertical wall of the step. The point where the horizontal velocity changes sign indicates the end of the recirculation area.

The length of the circulation area computed from Figure 6 is 6.79. This value compares very well (3.2%) with the experimental value reported by Kim et al. [59] (see Table 1). The FIC/FEM results are remarkably accurate in comparison with other results reported in the literature obtained using Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle K-\varepsilon }

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle K}

-tau turbulence models [60–63]. We note again that the FIC/FEM formulation does not include any additional turbulence terms.

Distribution of the horizontal velocity along the bottom line starting from the vertical wall of the step. The circle shows the end of the recirculation region.

[60]

[61]

[62]

[63]

7.2 Flow past a cylinder. Computation of the Strouhal unstability

Figure 7 shows the geometry for the analysis of the flow past a cylinder of unit diameter (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle D} ). A unit horizontal velocity is prescribed at the inlet boundary and at the two horizontal walls. Zero pressure is prescribed at the outlet boundary. The dimensions of the analysis domain are Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 27\times 27}

 units. The origin of the coordinate system has been sampled at the center of the cylinder located at a distance of 13.1 units from the entry wall. Zero velocity is prescribed at the cylinder wall. The kinematic viscosity is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \nu =0.01}

. Figure 8 shows the mesh of 91316 three-noded elements used for the computation. A detail of the mesh in the vicinity of the cylinder is also shown in Figure 8.

Flow past a cylinder. Mesh of 91316 three-noded triangles used for the computations.

secs.

The problem has been analyzed first for a value of the horizontal velocity at the entry of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle u_1=1}

giving a Reynolds number of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle Re =100}

. Figures 9 and 10 respectively show the velocity modulus contours and the velocity vectors for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle t=100} secs.

Figure 11 shows images of the trajectory of a substance over a band of 2.45 units transported at the entry across the flow for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle t=100} secs. The picture shows clearly the oscillatory nature of the flow.

Figure 12 shows the oscillations of the horizontal velocity at the point Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle A}

with coordinates (6.7, -1.02) for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle t=100}

secs. The Strouhal number computed from the shedding frequency Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle n}

as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle S={nD\over \vert {u}\vert }}
is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle S=0.1702}

. This number compares very well with the experimental result available in the literature (see Figure 13).

The same problem was analyzed for a value of the kinematic viscosity Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \nu =0.001}

giving Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle Re =1000}

. The same mesh of 91316 linear triangles of Figure 7 was used. Figures 14–16 show respectively the velocity modulus contours, the velocity vectors in the vicinity of the cylinder for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle t=100}

secs. and the trajectories of a substance transported across the flow.  Figure 17 finally  shows the oscillations of the horizontal velocity at  point Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle A}

. The computed value of the Strouhal number in this case was Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle S=0.2103} . This value again coincides well with the reported experimental data (see Figure 13).

It is a well known fact that for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle Re > 300}

the flow past a cylinder exhibits 3D features. In  [64] results from 2D and 3D computation were compared for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle Re=300}
and 800. While 3D features were observed even at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle Re =300}
and more so at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle Re =800}

, there were no large discrepances between the global flow parameters (such as drag, lift and Strouhal number) obtained from 2D and 3D computations. These conclusions justify the results of the 2D computations presented in the paper.

secs.

secs.

(6.7–1.02).

Flow past a cylinder. Experimental (thick line) and computed (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\blacklozenge } ) values of the Strouhal number Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): S in terms of the Reynolds number. Experimental values taken from [2].

secs.

secs.

secs.

(6.7–1.02).

8 CONCLUSIONS

The finite calculus (FIC) form of the fluid mechanics equations is a good starting point for deriving stabilized FEM for solving a variety of incompressible fluid flow problems. The matrix stabilization terms introduced by the FIC formulation here presented allow to obtain physically sound solutions in the presence of sharp gradients occuring for high Reynolds numbers without the need of introducing a turbulence model. Good numerical solutions have been obtained in the 2D examples solved with relatively coarse meshes for moderate and high values of the Reynolds number. These preliminary results reinforce our idea that the stabilization terms introduced by the FIC formulation suffice to provide good results for problems for which turbulence models are required using alternative numerical methods. These results also confirm the close link between the stabilized methods and turbulence models, which surely will be the object of much research in the near future.

ACKNOWLEDGEMENTS

The authors thank Prof. S.R. Idelsohn and C.A. Felippa for many useful discussions.

APPENDIX

The FIC momentum equations in two dimensions (2D) are obtained by expressing the balance of momentum along the horizontal and vertical directions in the finite domains shown in Figures A.1 and A.2, respectively.

The balance equation is written for each finite domain as

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle f_i}

includes the forces due to the stresses acting on the boundary of the balance domain and the body forces per unit area (Figures A.1 and A.2).

Expressing the values of the momentum and force terms at the end point of the balance domain in terms the values at an arbitrary point (such as the corner point A) using higher order Taylor expansions and retaining second order tems gives after some algebra 32 the FIC momentum equations along the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle i} th coordinate direction as

with

with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \sigma _{ij}= \tau _{ij}-p \delta _{ij}}

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \tau _{ij}}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle p}
are the deviatoric stresses and the pressure, respectively.

Note that distance Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle h_{12}}

is arbitrary when writting the balance of momentum along the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x_1}
direction. The same applies for the distance Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle h_{21}}
when deriving  the balance equation along the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x_2}
direction. Thus, in general, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle h_{12}\not = h_{21}}
and this explains the matrix form of the FIC momentum equations.

The FIC mass balance equation is obtained by invoking the balance of mass in the finite domain of Figure A.3

Finite domain where balance of mass is enforced

Expanding the values of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \rho u_i}

at the corner points in terms of the value at an arbitrary  point gives of the mass balance domain the FIC mass balance equation as 32

Note that a matrix form of the characteristic distances is not obtained in this case as the mass balance equation expresses the conservation of the mass in the whole domain ABCD of Figure A.3 with dimensions Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle h_1}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle h_2}

. Distances Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle h_1}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle h_2}
should be taken  in general different from distances Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle h_{ij}}
defining the domain where balance of momentum is enforced. In our computations we have however assumed that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle h_1 = h_{11}}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle h_2 = h_{22}}
for simplicity.

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