Method for computing turbulent flow using a near-wall eddy-viscosity formulation

Abstract

A technique that improves large-eddy simulation consists in replacing the instantaneous sub-grid scale eddy-viscosity (such as the dynamic Smagorinsky model eddy-viscosity) in the near-wall region with an eddy-viscosity computed from Reynolds Averaged Navier-Stokes eddy-viscosity and corrected dynamically using the resolved turbulent stress. The near-wall eddy-viscosity formulation is applied either with a wall stress model on coarse grids that do not resolve the wall or with wall-resolved grids coarsened in the wall-parallel directions. Reynolds averaged Navier-Stokes eddy-viscosity is computed either from a look-up table or from a simultaneous solution of a Reynolds Averaged Navier-Stokes turbulence model.

Description

DRAWINGS

Figures

FIG. 1 shows the conceptual application of the near-wall eddy-viscosity formulation to large-eddy simulation of turbulent flow around an airfoil.

FIG. 2 shows the details of the application of the near-wall eddy-viscosity in the context of wall-parallel coarsening.

FIG. 3 shows the details of the application of the near-wall eddy-viscosity in the context of wall modeling.

FIG. 4 shows the conceptual application of eddy-viscosity formulation to LES/RANS coupling in an internal flow.

REFERENCE NUMERALS

10	outer LES region	12	near-wall region
14	airfoil	16	LES region
18	RANS region

DETAILED DESCRIPTION

Preferred Embodiment—FIG. 1

The invented near-wall eddy-viscosity formulation has been developed as an ingredient of LES aimed at reducing its cost for turbulent flows around wings, blades and bluff bodies. FIG. 1 shows the conceptual application of the near-wall eddy-viscosity formulation to large-eddy simulation flow around an airfoil. The computational mesh (domain) around an airfoil 14 is split in two distinct regions: the outer LES region 10 and the near-wall region 12.

The invented formulation consists in replacing the instantaneous SGS eddy-viscosity, v_t^SGS, in the near-wall region of an LES with the invented near-wall eddy-viscosity, v_t^SGS,NW, defined with the following relation:

$\begin{array}{cc}{v}_t^{\mathrm{SGS},\mathrm{NW}}=v_t^{\mathrm{RANS}}+\stackrel{\_}{{\hat{u}}^{\prime }{\hat{v}}^{\prime }}/\frac{\stackrel{\_}{\hat{u}}}{y},& \left(1\right)\end{array}$

as illustrated in FIG. 1.

In equation (1) the term

$\stackrel{\_}{{\hat{u}}^{\prime }{\hat{v}}^{\prime }}$

is the resolved Reynolds shear stress, û′ and {circumflex over (v)}′ are the instantaneous streamwise and wall-normal velocity fluctuations, defined as

${\hat{u}}^{\prime }=\hat{u}-\stackrel{\_}{\hat{u}}$

and

${\hat{v}}^{\prime }=\hat{v}-\stackrel{\_}{\hat{v}},$

respectively.

$\frac{\stackrel{\_}{\hat{u}}}{y}$

is the wall-normal derivative of the average streamwise velocity,

$\stackrel{\_}{\hat{u}},$

and V_i^RANS is the RANS eddy-viscosity. The averaging operator ( ) is either time-averaging in general three-dimensional flows or plane-averaging in two-dimensional flows and spanwise-averaging in flows with one homogeneous direction.

The near-wall region is defined so that it comprises of all computational cells which are at a distance to the solid surface smaller than y^NW (for example, for an airfoil y^NW is typically less than 20 percent of the airfoil cord).

RANS eddy-viscosity is computed either from a look-up table or from a simultaneous solution of a RANS turbulence model.

OPERATION

Preferred Embodiment—FIGS. 2 and 3

The near-wall eddy-viscosity formulation is applied either with a wall stress model on coarse grids that do not resolve the wall or with wall-resolved grids coarsened in the wall-parallel directions.

The application of the near-wall formulation with the wall-parallel coarsening is shown in FIG. 2. With wall-parallel coarsening, the values of the non-dimensional distance to the nearest wall, y⁺=u_τy/v (where u_τ is the friction velocity), are usually lower than 1 for the wall-adjacent cells (the wall-adjacent cells are defined here as the cells that have a common side with a wall). The no-slip boundary condition is applied for the velocity, û_W=0. The near-wall eddy-viscosity, v_t^SGS,NW, is applied in the near-wall region that typically covers up to 15 computational cells in the wall-normal direction. The sub-grid scale eddy-viscosity, v_t^SGS, is applied elsewhere. The computational savings are achieved by coarsening the grid in the wall-parallel directions (It is possible to use grids with the non-dimensional spacing in the streamwise, Δx⁺=Δxu_τ/v, and spanwise directions, Δz⁺=Δzu_τ/v, of approximately 50 to 100).

The application of the near-wall formulation with a wall stress model on coarse grids is illustrated in FIG. 3. The wall stress model replaces the no-slip boundary condition, û_W=0, used with the wall-parallel coarsening, with a boundary condition on the wall stress, τ_w. The corresponding y⁺ values for the wall-adjacent cells are usually in the so-called logarithmic region of the boundary layer, i.e. y⁺>30, thus making the computational grids significantly coarser and more uniform. When the near-wall eddy-viscosity, v_i^SGS,NW, is used with a wall stress model, it is only applied in the wall-adjacent cells.

Alternative Embodiments

The proposed near-wall eddy-viscosity formulation can be also be computed using

$\stackrel{\_}{{\hat{u}}_i^{\prime }{\hat{u}}_j^{\prime }},$

the resolved Reynolds stress tensor, in the least square sense:

$\begin{array}{cc}{v}_t^{\mathrm{SGS},\mathrm{NW}}=v_t^{\mathrm{RANS}}+\sum _i\sum _j\stackrel{\_}{{\hat{u}}_i^{\prime }{\hat{u}}_j^{\prime }}{\stackrel{\_}{\hat{S}}}_{\mathrm{ij}}/\left(2{\stackrel{\_}{\hat{S}}}^2\right)& \left(2\right)\end{array}$

where û_i′ are the velocity fluctuations,

${\stackrel{\_}{\hat{S}}}_{\mathrm{ij}}=\frac{1}{2}\left(\frac{\partial {\stackrel{\_}{\hat{u}}}_i}{\partial x_j}+\frac{\partial {\stackrel{\_}{\hat{u}}}_j}{\partial x_i}\right)$

is the strain rate tensor computed for the average velocity,

${\stackrel{\_}{\hat{u}}}_i,\mathrm{and}{\stackrel{\_}{\hat{S}}}^2=\sum _i\sum _j{\stackrel{\_}{\hat{S}}}_{\mathrm{ij}}{\stackrel{\_}{\hat{S}}}_{\mathrm{ij}}.$

Another variation of the method is to use an approximation for the instantaneous sub-grid scale stress in the near-wall region directly:

$\begin{array}{cc}{\left(v_t^{\mathrm{SGS}}\frac{\stackrel{\_}{\hat{u}}}{y}\right)}_{\mathrm{NW}}=\left(v_t^{\mathrm{RANS}}\frac{\stackrel{\_}{u}}{y}\right)+\stackrel{\_}{{\hat{u}}^{\prime }{\hat{v}}^{\prime }}& \left(3\right)\end{array}$

Additional Embodiments

The eddy-viscosity formulation can also be used for LES/RANS coupling in internal flows, such as presented in FIG. 4, to provide the inflow turbulence data for the RANS computational region:

$\begin{array}{cc}{v}_t^{\mathrm{RANS}}=\stackrel{\_}{v_t^{\mathrm{SGS}}}-\sum _i\sum _j\stackrel{\_}{{\hat{u}}_i^{\prime }{\hat{u}}_j^{\prime }}{\stackrel{\_}{\hat{S}}}_{\mathrm{ij}}/\left(2{\stackrel{\_}{\hat{S}}}^2\right)& \left(4\right)\end{array}$

where û_i′ are the velocity fluctuations,

${\stackrel{\_}{\hat{S}}}_{\mathrm{ij}}=\frac{1}{2}\left(\frac{\partial {\stackrel{\_}{\hat{u}}}_i}{\partial x_j}+\frac{\partial {\stackrel{\_}{\hat{u}}}_j}{\partial x_i}\right)$

is the strain rate tensor computed for the average velocity,

${\stackrel{\_}{\hat{u}}}_i,\mathrm{and}{\stackrel{\_}{\hat{S}}}^2=\sum _i\sum _j{\stackrel{\_}{\hat{S}}}_{\mathrm{ij}}{\stackrel{\_}{\hat{S}}}_{\mathrm{ij}}.\stackrel{\_}{v_t^{\mathrm{SGS}}}$

is the average sub-grid scale eddy-viscosity. The RANS eddy-viscosity, v_t^RANS, is then used to compute turbulence variables, for example, in k-ω model it is used to compute ω if k is computed as

$k=\frac{1}{2}\sum _i\stackrel{\_}{{\hat{u}}_i^{\prime }{\hat{u}}_j^{\prime }}.$

CONCLUSIONS, RAMIFICATIONS, AND SCOPE

From the description above, a number of advantages of our invention become evident. Our invention makes it possible to perform accurate large-eddy simulation of high Reynolds flows with the currently existing computer hardware. Compared to other near-wall formulations, it has the advantage of being consistent with wall-resolved large-eddy simulation, that it is simple, easy to implement and that it adds negligible extra computational cost. The near-wall eddy-viscosity has been successfully tested flows at high Reynolds numbers, as presented in Kalitzin, G., Templeton, J. A., and Medic, G. (2006), “A near-wall eddy-viscosity formulation for LES”, Lecture Notes in Computational Science and Engineering Vol. 56, Springer-Verlag: the computed results are superior to results from the large-eddy simulations that do not use our near-wall eddy-viscosity.

Accordingly, the reader will see that the near-wall eddy-viscosity formulation of this invention can significantly reduce the computational cost of predicting high Reynolds number turbulent flows around wings, blades and bluff bodies (such as cars) by improving the accuracy of the LES on coarse near-wall grids. In addition, several advantages of the present invention are in that:

• • it provides a model that automatically adapts to the computational grid that is used in the simulation;
  • it provides a model that automatically adapts to flow Reynolds number;
  • it provides a model that, compared to other near-wall formulations for LES, has the advantage of being consistent with full-blown wall-resolved LES;
  • it provides a model that is simple and easy to implement in a variety of computational codes; and
  • it provides a model that adds negligible extra computational cost to performing LES.

Although the description above contains many specificities, these should not be construed as limiting the scope of the invention but as merely providing illustrations of some of the presently preferred formulations of this invention. For example, the eddy-viscosity formulation can be used as an ingredient in coupling general RANS and LES computational codes beyond the near-wall region.

Thus, the scope of this invention should be determined by the appended claims and their legal equivalents, rather than the examples given.

Claims

1. A method to compute turbulent flow, comprising of the following steps: reading object geometry for providing points on a surface of an object; establishing a computational mesh around the said object; marking the computational cells in a region near the surface of the said object; calculating turbulent flow field data in the entire computational mesh by solving the filtered Navier-Stokes equations using large-eddy simulation; substituting the sub-grid scale eddy-viscosity in the said region near the surface of the said object with a near-wall eddy-viscosity; computing the near-wall eddy-viscosity from a Reynolds Averaged Navier-Stokes eddy-viscosity and correcting it dynamically using the resolved turbulent fluctuations;

2. The method of claim 1, wherein the near-wall eddy-viscosity formulation comprises of computing an eddy-viscosity from a Reynolds Averaged Navier-Stokes eddy-viscosity corrected with the ratio of an average of the resolved Reynolds shear stress over the wall normal derivative of the average velocity;

3. The method of claim 2, wherein said step of computing the Reynolds Averaged Navier-Stokes eddy-viscosity comprises of using tabulated data of eddy-viscosity pre-computed for attached zero pressure gradient boundary layers;

4. The method of claim 2, wherein said step of computing the Reynolds Averaged Navier Stokes eddy-viscosity comprises of solving simultaneously a Reynolds Averaged Navier Stokes turbulence model;

5. The method of claim 2, wherein the average of the resolved Reynolds shear stress is computed either as a spanwise average, as a span- and streamwise average or as a time average;

6. The method of claim 1, wherein the near-wall eddy-viscosity formulation is generalized with the ratio of the product of the average of the resolved Reynolds stress tensor with the average strain rate tensor over the square of the average strain rate tensor multiplied by two;

7. The method of claim 1, wherein in the said near-wall region the sub-grid scale stress is computed from the Reynolds Averaged Navier-Stokes Reynolds stress corrected with the resolved Reynolds stress;

8. The method of claim 1, wherein computational mesh is coarsened only in wall-parallel directions and the cell center of the wall adjacent cells is located in the viscous sub-layer;

9. The method of claim 8, wherein the no-slip boundary conditions are used for the velocity;

10. The method of claim 1, wherein computational mesh is coarsened in wall-parallel and wall normal directions;

11. The method of claim 10, wherein a wall stress boundary condition is used for the velocity;

12. The method of claim 11, wherein the wall stress is obtained from tabulated data of non-dimensionalized velocity pre-computed for attached zero pressure gradient boundary layers;

13. The method of claim 11, wherein the wall stress is obtained from a simultaneous Reynolds Averaged Navier-Stokes simulation on a Reynolds Averaged Navier-Stokes grid.

14. A method to compute turbulent flow, comprising of the following steps: reading object geometry for providing points on a surface of an object: establishing a computational mesh around the said object; dividing the computational mesh in two distinct regions; calculating turbulent flow field data in one of the said computational region by solving the filtered Navier-Stokes equations using large-eddy simulation; calculating turbulent flow field data in the other of the said computational regions by solving the Reynolds Averaged Navier-Stokes equations; using the invented eddy-viscosity formulation to exchange turbulence data between the two computational regions.