Revision as of 18:56, 8 December 2009

2D Periodic Hill

Underlying Flow Regime 3-30

Test Case

Brief Description of the Test Case Studied

The measures of the geometry introduced by Mellen et al. (2000) relate to the hill height h. The hill constricts the channel height of 3.036h by about one third, whereas the inter hill distance is 9h. The contour of the 3.857h long two-dimensional hill is described by the following six polynomials.

$x\;\epsilon \;\left[0;\;0.3214h\right];\quad y(x)=\min(1;1+0\cdot x+2.420\cdot 10^{-4}\cdot x^{2}-7.588\cdot 10^{-5}\cdot x^{3});$ $x\;\epsilon \;\left[0.3215h;\;0.5h\right];\quad y(x)=0.8955+3.484\cdot 10^{-2}\cdot x-3.629\cdot 10^{-3}\cdot x^{2}+6.749\cdot 10^{-5}\cdot x^{3};$ $x\;\epsilon \;\left[0.5h;\;0.7143h\right];\quad y(x)=0.9213+2.931\cdot 10^{-2}\cdot x-3.234\cdot 10^{-3}\cdot x^{2}+5.809\cdot 10^{-5}\cdot x^{3};$ $x\;\epsilon \;\left[0.7144h;\;1.071h\right];\quad y(x)=1.445-4.927\cdot 10^{-2}\cdot x+6.950\cdot 10^{-4}\cdot x^{2}-7.394\cdot 10^{-6}\cdot x^{3};$ $x\;\epsilon \;\left[1.071h;\;1.429h\right];\quad y(x)=0.6401+3.123\cdot 10^{-2}\cdot x-1.988\cdot 10^{-3}\cdot x^{2}+2.242\cdot 10^{-5}\cdot x^{3};$ $x\;\epsilon \;\left[1.429h;\;1.929h\right];\quad y(x)=\max(0;2.0139-7.180\cdot 10^{-2}\cdot x+5.875\cdot 10^{-4}\cdot x^{2}+9.553\cdot 10^{-7}\cdot x^{3});$

At x/h=0 the hill height is maximal, whereas the boundary is flat in the range between x/h=1.929 and x/h=7.071. Between x/h=7.071 and x/h=9 the contour follows the above equations but the hill geometry is mirrored at x/h = 4.5. Besides the geometry, the following figure shows streamlines at a Reynolds number, that is based on the hill height h and the bulk velocity above the crest, of 5,600 [Rapp 2009)].

$Re={\frac {u_{b}h}{\nu }}$

$u_{b}={\frac {1}{2.035}}\int _{h}^{3.035h}u\left(y\right)dy$

The mean flow separates at the curved hill crown. In the wake of the hill the fluid recirculates before it attaches naturally at about x/h=4.5.

Experimental Setup

A water channel has been set up in the Laboratory for Hydromechanics of the Technische Universität München to investigate the flow experimentally. In total 10 hills with a height of 50 mm were built into the rectangular channel to accomplish periodicity whilst the measurement range lies between hills seven and eight. To achieve homogeneity in the spanwise direction an extent of 18 hill heights was appointed. The following figure sketches the experimental setup.

The 2D PIV measurements were undertaken between hills seven and eight - and to investigate the periodicity of the flow - between the hill pair six and seven through vertical laser light sheets. The homogeneity in the spanwise direction was controlled by 2D PIV measurements in horizontal planes. The PIV field data was thoroughly validated through 1D LDA measurements. Experiments were done at four Reynolds numbers: Re=5,600; Re=10,600; Re=19,000 and Re=37,000.

CFD Methods

The numerical part of the present study relies on two completely independent codes based on either curvilinear body-fitted grids with a colocated variable arrangement or Cartesian non-uniform grids using a staggered configuration. The objective is to present highly reliable results obtained by carefully cross-checking between the outcome of both numerical schemes and additional experimental data. Afterwards the investigations concentrate on the physical aspects of the flow considered.

In the following, both codes are described briefly.

Finite-volume code LESOCC

LESOCC solves the (filtered) Navier-Stokes equations based on a three-dimensional finite-volume method for arbitrary non-orthogonal and non-staggered block-structured grids (see, e.g.\ Fig. ??). The spatial discretization of all fluxes is based on central differences of second-order accuracy. Time advancement is performed by a predictor--corrector scheme. A low-storage multi-stage Runge-Kutta method (three sub-steps, second-order accuracy) is applied for integrating the momentum equations in the predictor step. Within the corrector step the Poisson equation for the pressure correction is solved implicitly by the incomplete LU decomposition method. Explicit time marching works well for DNS and LES with small time steps which are necessary to resolve turbulence motion in time. In order to ensure the coupling of pressure and velocity fields on non-staggered grids, the momentum interpolation technique is used. For modeling the non-resolvable subgrid scales, two different models are implemented, namely the well-known Smagorinsky model (1963) with Van Driest damping near solid walls and the dynamic approach with a Smagorinsky base model proposed by Germano et al. (1991) and modified by Lilly (1992). In order to stabilize the dynamic model, averaging of the numerator and the denominator in the relation for the determination of the Smagorinsky value was carried out in the spanwise homogeneous direction and also in time using a recursive digital low-pass filter (Breuer and Rodi 1996, Breuer 2002). The code and the implemented SGS models were validated on a variety of different test cases. For more information on this issue, please refer to Breuer and Rodi (1996), Breuer (1998, 2000, 2002).

Finite-volume code MGLET

MGLET is based on a finite-volume formulation for non-uniform Cartesian grids with a staggered arrangement of the spatially filtered variables (see, e.g.\ Fig. ??). The spatial discretization of the convective and diffusive fluxes is based on second-order central differences. The momentum equations are advanced in time by a fractional time stepping using either an explicit second-order central leapfrog scheme or a third-order Runge-Kutta scheme. For the solution of the Poisson equation for the pressure the ``Strongly Implicit Procedure (SIP) is implemented. For the representation of the hill geometry in the Cartesian grid an immersed boundary technique is used. All Cartesian cells lying inside the body are excluded from the computation. The excluded cells are determined by the intersection of the hill geometry with the Cartesian cells. The geometry of the hills is represented by a triangle mesh. The immersed boundary technique provides a smooth representation of the body surface in the Cartesian mesh by using third-order least squares interpolation for the interface cells (Peller 2006). This method prevents instabilities which are present in high-order Lagrange interpolation schemes. The code is used for DNS and LES simulations. It has been shown by several authors that second-order accuracy can be sufficient for DNS of flows provided the grid resolution is sufficient (Manhart and Friedrich 2002, Peller et al. 2006).

Boundary conditions and simulation parameters

Since the grid resolution in the vicinity of the wall is sufficient to resolve the viscous sublayer, the no-slip and impermeability boundary condition is used at the wall in both codes. The flow is assumed to be periodic in the streamwise direction and thus periodic boundary conditions are applied. Similar to the turbulent plane channel flow case the non-periodic behavior of the pressure distribution can be accounted for by adding the mean pressure gradient as a source term to the momentum equation in streamwise direction. Two alternatives exist. Either the pressure gradient is fixed which might lead to an unintentional mass flux in the configuration or the mass flux is kept constant which requires to adjust the mean pressure gradient in time. Since a fixed Reynolds number can only be guaranteed by a fixed mass flux, the second option is chosen here.

Furthermore, the flow is assumed to be homogeneous in spanwise direction and periodic boundary conditions are applied, too. For that purpose the use of an adequate domain size in the spanwise direction is of major importance in order to obtain reliable and physically reasonable results. To assure this criterion the two-point correlations in the spanwise direction have to vanish in the half-width of the domain size chosen. Based on the investigations by Mellen et al. (2000) a spanwise extension of the computational domain of L_z = 4.5 h is used in all computations presented. It represents a well-balanced compromise between spanwise extension and spanwise resolution.

The Table summarizes the most important parameters of the simulations available. N_{tot} denotes the total number of grid points used; the corresponding number of control volumes is slightly lower. Although a direct comparison of the number of grid points used by LESOCC and MGLET in one x-y plane is not reasonable, at least the number of points equidistantly distributed in the spanwise direction, N_{span}, can be compared.

The dimensionless time-step size $\Delta t$ is also tabulated in Table~\ref{tab:simtab}, where the time is normalized by the ratio of the hill height $h$ and the bulk velocity ${U_B}$ taken at the crest of the hill. To reduce statistical errors due to insufficient sampling to a reasonable minimum, the flow field was averaged in spanwise direction and in time over a long period of $\Delta T_{avg}$ which is also given in Table~\ref{tab:simtab}. Partially $\Delta T_{avg}$ covers a time interval of about ${140}$ flow-through times.

Contributed by: (*) Christoph Rapp, (**) Michael Breuer — (*) Technische Universitat München, (**) Helmut-Schmidt Universität Hamburg