# Boundary layer flow and dispersion over isolated hills and valleys

Application Challenge 5-05 © copyright ERCOFTAC 2004

## Overview of CFD Simulations

AC describe the computational techniques; a brief introduction is given above. Further details are now summarised but it should be noted there are no available data files containing the computational results.

## Simulation Case

Solution strategy

This is outlined in the Introduction.

Computational Domain

As noted earlier, the (2-D) solution domain extended to x=±40H and z=13.7H. (The aspect ratio of the experimental hills in the tunnel was thought large enough to give essentially 2D behaviour on the tunnel centre-line).

Boundary Conditions

CA define the upstream boundary conditions imposed in all calculations – essentially a rough-wall log-law formulation for the mean velocity and a turbulence kinetic energy falling quadratically with height. Each turbulence model – whether k-ε or any of the variants used, also required an inlet ε-profile, which was derived in the usual way from the energy profile (see CA for details). Zero vertical gradients of k and ε were imposed at the upper boundary and U was set at the constant, measured Uo at that height. A small mass flux through the upper boundary was allowed so as to satisfy continuity (given the small boundary layer growth through the domain). At the downstream boundary zero-gradient conditions were imposed on all variables. A little more detail on the rough-wall conditions is necessary. In the axial momentum equation an effective viscosity (giving the wall stress) is enforced as ${\displaystyle \nu _{t,eff}=(\kappa C_{\mu }^{3/4}k_{p}^{3/2}z_{p})/\ln(z_{p}/z_{0})}$ where ${\displaystyle z_{0}}$ is the roughness length and ${\displaystyle z_{p}}$ is the distance from the wall to the nearest mesh point. (Note that this implies that the velocity scale is taken as ${\displaystyle C_{\mu }^{1/4}k_{p}^{1/2}}$ , rather than the usual ${\displaystyle U_{\tau }}$; the latter is only correct in equilibrium conditions and is often very inappropriate – e.g. at a stagnation point). In the k-equation, expressions for the cell-averaged production and dissipation are used for the near-wall control volume; these are derived in terms of ${\displaystyle K_{p}}$ and ${\displaystyle U_{p}}$ and are only equal in strictly equilibrium conditions. In the dissipation equation, the dissipation at the near-wall point is set to be given by ${\displaystyle \epsilon =C_{\mu }^{3/4}k_{p}^{3/2}/\kappa k_{p}}$. This formulation is undoubtedly better than that employed in most commercial packages (it is certainly more realistic in regions of very small wall stress and high turbulence levels) but is not claimed to be perfect. No straightforward alternative is available for flows, like the present case, in which there are regions where no log-law exists.

Application of Physical Models

In addition to the standard k-ε model, CA used variants: one in which a slightly modified version of the Leschziner & Rodi (1981) model, which includes a curvature correction, and another, suggested by the same authors, which embodies a ‘dissipation modification’ – designed to correct the inappropriate response of the standard model to strong streamwise strains. The model constants (given in CA) were consistent with constant flux, surface layer similarity (log-law) profiles. To compute the scalar field, the computed velocity and turbulence fields were first interpolated onto a second grid, refined near the source location. This required care in ensuring overall global conservation of material. An embedded analytical domain was used to specify concentrations near the source, via the exact solution of the 3D advection-diffusion equation with constant coefficients. Then the full 3D scalar transport equation was solved using the same techniques as for the flow field computations.

Numerical Accuracy

A number of computations were undertaken (using grids ranging from 60 x 40 to 200 x 160 nodes) to check that solutions (on the ‘standard’ grid of 100 x 80) were grid independent. Again, details are provided in CA, where it is demonstrated that the solutions were, indeed, grid independent – a result no doubt of the superior accuracy of the Van Leer advection scheme used. The minimum control volume dimension (near the hill summit) was 0.1H x 0.004H. Note that the usual smooth-wall ‘y+-condition’ was not relevant for this rough-surface flow. But it was important to check that the code successfully computed the continuing development of the boundary layer in the absence of the hill; this is the best way to ensure self-consistency of the model. In all cases, the (iterative) computations were continued until normalised residuals in each of the equations fell to sufficiently small values (typically smaller than 10-4). By that stage all results were steady, with no further changes apparent on further iterations.

CFD Results

Currently the CFD data are not available in the ERCOFTAC database but typical results are shown in CA.

## References

As in Experimental References, plus:

Ying, R., Canuto, V.M. & Ypma, R.M. (1994). Numerical simulation of flow data over two-dimensional hills, Bound.Layer Met., 70, 401-427.

Ying, R. & Canuto, V.M. (1996). Turbulence modelling over two-dimensional hills using an algebraic Reynolds stress expression, Bound.Layer Met., 77, 69-99.

Ying, R. & Canuto, V.M. (1997). Numerical simulation of flow over two-dimensional hills using a second-order turbulence closure model. Bound.Layer Met., 85, 447-474

Leschziner, M.A. & Rodi, W. (1981). Calculation of annular and twin parallel jets using various discretisation schemes and turbulence model variations, J. Fluids Eng.. 103, 352-360.

Note that the Ying & Canuto work does not include dispersion calculations.