Periodic Hill Flow (RANS and Hybrid RANS/LES)

The periodicHill case models turbulent flow over a periodic array of two-dimensional hills. The benchmark configuration and reference Large Eddy Simulation (LES) results are adopted from Fröhlich et al. (Fröhlich et al., 2005). The adverse pressure gradient downstream of each hill produces flow separation, a recirculation region, and subsequent reattachment along the lower wall. The case verifies the $var element = document.getElementById("moose-equation-ee5f4958-ca2d-4f1c-965a-2aeefd12001e");katex.render("k", element, {displayMode:false,throwOnError:false});$ - $var element = document.getElementById("moose-equation-f37a8374-5d24-4f8a-8318-d0a04f99cc71");katex.render("\\tau", element, {displayMode:false,throwOnError:false});$ Shear-stress Transport (SST) model and its Delayed Detached Eddy Simulation (DDES) and Improved Delayed Detached Eddy Simulation (IDDES) variants in NekRS. The $var element = document.getElementById("moose-equation-bba029df-358d-440c-be4d-a1804d7e1220");katex.render("k", element, {displayMode:false,throwOnError:false});$ - $var element = document.getElementById("moose-equation-aab28fda-67aa-4a07-b869-bc2bf10667be");katex.render("\\tau", element, {displayMode:false,throwOnError:false});$ SST formulation is described by Tomboulides et al. (Tomboulides et al., 2025), while the hybrid RANS/LES formulations are discussed by Chang et al. (Chang et al., 2023).

Computational domain

The hill height $var element = document.getElementById("moose-equation-e750de38-1bae-4d4f-b816-cb65d93c9966");katex.render("H=1", element, {displayMode:false,throwOnError:false});$ is used as the reference length. The domain dimensions are

(1)var element = document.getElementById("moose-equation-4aae0284-122b-43c2-87e4-63e670e949d4");katex.render("L_x=9H,\\qquad L_y=3.035H,\\qquad L_z=4.5H.", element, {displayMode:true,throwOnError:false});

The lower boundary consists of a periodic hill profile with a width parameter $var element = document.getElementById("moose-equation-90ebe6c0-03b7-4397-8a69-d253e1b9fa6e");katex.render("W=1.929H", element, {displayMode:false,throwOnError:false});$ , while the upper boundary is flat. The domain is discretized using $var element = document.getElementById("moose-equation-d9874ac9-45cd-4be9-a640-e3880890d6aa");katex.render("12\\times8\\times9", element, {displayMode:false,throwOnError:false});$ hexahedral elements in the streamwise, wall-normal, and spanwise directions, respectively, for a total of 864 elements. A polynomial order of five is used within each element. The mesh is stretched in the streamwise and wall-normal directions to increase resolution near the hill and within the separated shear layer.

Periodic boundary conditions are imposed in the streamwise $var element = document.getElementById("moose-equation-fe3137b1-0e08-4374-9418-f1584c23ff91");katex.render("x", element, {displayMode:false,throwOnError:false});$ and spanwise $var element = document.getElementById("moose-equation-289c2cbe-0ce0-45b2-846a-011c7cdf02b0");katex.render("z", element, {displayMode:false,throwOnError:false});$ directions. No-slip boundary conditions are imposed on the lower hill surface and the upper wall. The turbulent quantities $var element = document.getElementById("moose-equation-c8bdfaf7-69d6-40a0-99f4-1aa21b53eedc");katex.render("k", element, {displayMode:false,throwOnError:false});$ and $var element = document.getElementById("moose-equation-e81e8426-91ee-482a-9753-926d0df7a079");katex.render("\\tau", element, {displayMode:false,throwOnError:false});$ are assigned zero Dirichlet boundary conditions at the walls.

Flow parameters

The bulk velocity, fluid density, and hill height are nondimensionalized as

(2)var element = document.getElementById("moose-equation-d3ddf022-5566-4a95-8603-a43138b4dbc4");katex.render("U_b=1,\\qquad \\rho=1,\\qquad H=1.", element, {displayMode:true,throwOnError:false});

The Reynolds number based on the bulk velocity and hill height is

(3)var element = document.getElementById("moose-equation-c0341070-cd1d-445d-8e2b-67bcefaa191b");katex.render("Re_H = \\frac{\\rho U_bH}{\\mu} = \\frac{U_bH}{\\nu} = 10595,", element, {displayMode:true,throwOnError:false});

where $var element = document.getElementById("moose-equation-b323ab01-710b-44fc-a752-63a5c4ea85e8");katex.render("\\mu", element, {displayMode:false,throwOnError:false});$ and $var element = document.getElementById("moose-equation-98833601-e8a7-4112-b800-12fc819a171e");katex.render("\\nu", element, {displayMode:false,throwOnError:false});$ are the dynamic and kinematic viscosities, respectively. The corresponding nondimensional molecular viscosity is

(4)var element = document.getElementById("moose-equation-3d52ce5d-7fdc-4716-944a-fbde2c215f7a");katex.render("\\nu=\\frac{1}{10595}.", element, {displayMode:true,throwOnError:false});

A constant bulk flow rate is maintained in the streamwise direction throughout the simulation.

Test Case

Three simulations are used to verify the turbulence-model implementations. Each mode restarts from a statistically developed flow field and advances the solution for 0.5 nondimensional time units. During this interval, the three velocity components are accumulated in time to obtain the mean velocity field.

The turbulence models and qualification tolerances for the three test cases are summarized in Table 1.

Table 1: Periodic-hill test cases and skin-friction error tolerances.

Test Case	Turbulence model	Restart field	Error tolerance
1	$var element = document.getElementById("moose-equation-92fafe21-8a17-415b-b6d9-220b882a4304");katex.render("k", element, {displayMode:false,throwOnError:false});$ - $var element = document.getElementById("moose-equation-baafda81-3df9-4a22-a946-19fad03157ff");katex.render("\\tau", element, {displayMode:false,throwOnError:false});$ SST	`sst.fld`	$var element = document.getElementById("moose-equation-f5dd57bd-124f-4dea-9fc7-9ac7e47c10c1");katex.render("1.04\\times10^{-2}", element, {displayMode:false,throwOnError:false});$
2	$var element = document.getElementById("moose-equation-0ecc82f9-cf9a-4ad3-a5ce-3da266cf3066");katex.render("k", element, {displayMode:false,throwOnError:false});$ - $var element = document.getElementById("moose-equation-bc486683-571b-4980-8f07-bdc2d27bf30e");katex.render("\\tau", element, {displayMode:false,throwOnError:false});$ SST DDES	`ddes.fld`	$var element = document.getElementById("moose-equation-bc5214ac-c346-4d10-b92c-315210425934");katex.render("1.14\\times10^{-2}", element, {displayMode:false,throwOnError:false});$
3	$var element = document.getElementById("moose-equation-f92d115f-76dc-48db-bfc2-543f726febbb");katex.render("k", element, {displayMode:false,throwOnError:false});$ - $var element = document.getElementById("moose-equation-86d0bf61-2176-48e0-a030-cb892dd6d90c");katex.render("\\tau", element, {displayMode:false,throwOnError:false});$ SST IDDES	`iddes.fld`	$var element = document.getElementById("moose-equation-6ec28da0-099b-471d-9530-075a231fa3f7");katex.render("1.07\\times10^{-2}", element, {displayMode:false,throwOnError:false});$

Verification criteria

The time-averaged velocity field is used to compute the mean strain-rate tensor and viscous shear stress along the periodic-hill wall. The local skin-friction coefficient is defined as

(5)var element = document.getElementById("moose-equation-ef41f0df-4674-4638-b965-9e1324dff93d");katex.render("C_f(x) = \\frac{2\\tau_w(x)}{\\rho U_b^2},", element, {displayMode:true,throwOnError:false});

where $var element = document.getElementById("moose-equation-9f5842c3-af4b-4768-af0c-931bfe04d952");katex.render("\\tau_w", element, {displayMode:false,throwOnError:false});$ is the viscous wall shear stress. Because $var element = document.getElementById("moose-equation-40b6a434-34af-4273-a7fb-18e646fbe1f6");katex.render("\\rho=1", element, {displayMode:false,throwOnError:false});$ and $var element = document.getElementById("moose-equation-67b9e1ec-4a47-458f-b0de-7ccb2754afa8");katex.render("U_b=1", element, {displayMode:false,throwOnError:false});$ for this case, the coefficient evaluated by the test reduces to

(6)var element = document.getElementById("moose-equation-bdcd1859-76eb-435c-a492-976424c9eb2b");katex.render("C_f(x)=2\\tau_w(x).", element, {displayMode:true,throwOnError:false});

The computed skin-friction coefficient is compared with the reference LES results of Fröhlich et al. (Fröhlich et al., 2005). The test uses the corresponding digitized skin-friction data available through the NASA Turbulence Modeling Resource. The reference coefficient is linearly interpolated onto the surface quadrature points of the NekRS mesh. The pointwise absolute error is

(7)var element = document.getElementById("moose-equation-b1caa1ea-e6e1-485e-a3e8-b5941106fb01");katex.render("e_{C_f}(x) = \\left| C_f(x)-C_{f,\\mathrm{ref}}(x) \\right|.", element, {displayMode:true,throwOnError:false});

The test qualification metric is the wall-area-averaged absolute error,

(8)var element = document.getElementById("moose-equation-7aa8c651-5e78-4cb9-9885-f73ab7f59056");katex.render("\\varepsilon_{C_f} = \\frac{ \\displaystyle\\int_{\\Gamma_w} \\left| C_f-C_{f,\\mathrm{ref}} \\right|\\,dA }{ \\displaystyle\\int_{\\Gamma_w}dA },", element, {displayMode:true,throwOnError:false});

where $var element = document.getElementById("moose-equation-f6410cc3-b6f8-4edf-af8a-ccf3c85f49a8");katex.render("\\Gamma_w", element, {displayMode:false,throwOnError:false});$ denotes the periodic-hill wall. Each case passes when $var element = document.getElementById("moose-equation-248091a0-0acb-42c7-9c05-9a1f659e5df7");katex.render("\\varepsilon_{C_f}", element, {displayMode:false,throwOnError:false});$ is below its corresponding tolerance in Table 1.

References

Kyoungsik Chang, Byeongcheon Kim, Nadish Saini, Ananias Tomboulides, Paul Fischer, and Misun Min. Evaluation of iddes in nek5000 for predicting the flow over periodic hills. Technical Report, Argonne National Laboratory (ANL), Argonne, IL (United States), 2023.

BibTeX

@techreport{chang2023evaluation,
    author = "Chang, Kyoungsik and Kim, Byeongcheon and Saini, Nadish and Tomboulides, Ananias and Fischer, Paul and Min, Misun",
    title = "Evaluation of IDDES in Nek5000 for predicting the flow over periodic hills",
    year = "2023",
    institution = "Argonne National Laboratory (ANL), Argonne, IL (United States)"
}

RIS

TY  - RPRT
AU  - Chang, Kyoungsik
AU  - Kim, Byeongcheon
AU  - Saini, Nadish
AU  - Tomboulides, Ananias
AU  - Fischer, Paul
AU  - Min, Misun
TI  - Evaluation of IDDES in Nek5000 for predicting the flow over periodic hills
PY  - 2023
ER  -

Plain Text

Kyoungsik Chang, Byeongcheon Kim, Nadish Saini, Ananias Tomboulides, Paul Fischer, and Misun Min. Evaluation of iddes in nek5000 for predicting the flow over periodic hills. Technical Report, Argonne National Laboratory (ANL), Argonne, IL (United States), 2023.

Jochen Fröhlich, Christopher P Mellen, Wolfgang Rodi, Lionel Temmerman, and Michael A Leschziner. Highly resolved large-eddy simulation of separated flow in a channel with streamwise periodic constrictions. Journal of Fluid Mechanics, 526:19–66, 2005.

BibTeX

@article{frohlich2005highly,
    author = {Fr{\"o}hlich, Jochen and Mellen, Christopher P and Rodi, Wolfgang and Temmerman, Lionel and Leschziner, Michael A},
    title = "Highly resolved large-eddy simulation of separated flow in a channel with streamwise periodic constrictions",
    journal = "Journal of Fluid Mechanics",
    volume = "526",
    pages = "19--66",
    year = "2005",
    publisher = "Cambridge University Press"
}

RIS

TY  - JOUR
AU  - Fröhlich, Jochen
AU  - Mellen, Christopher P
AU  - Rodi, Wolfgang
AU  - Temmerman, Lionel
AU  - Leschziner, Michael A
TI  - Highly resolved large-eddy simulation of separated flow in a channel with streamwise periodic constrictions
JO  - Journal of Fluid Mechanics
PY  - 2005
VL  - 526
PB  - Cambridge University Press
SP  - 19
EP  - 66
ER  -

Plain Text

Jochen Fröhlich, Christopher P Mellen, Wolfgang Rodi, Lionel Temmerman, and Michael A Leschziner. Highly resolved large-eddy simulation of separated flow in a channel with streamwise periodic constrictions. Journal of Fluid Mechanics, 526:19–66, 2005.

Ananias Tomboulides, Nadish Saini, DR Shaver, AV Obabko, Haomin Yuan, Elia Merzari, and PF Fischer. A robust spectral element implementation of the k-

var element = document.getElementById("moose-equation-2ef03f27-64ba-4d89-99ce-683b13593d80");katex.render("\\tau", element, {displayMode:false,throwOnError:false});

rans model in nek5000/nekrs. International Journal of Heat and Fluid Flow, 112:109679, 2025.

BibTeX

@article{tomboulides2025robust,
    author = "Tomboulides, Ananias and Saini, Nadish and Shaver, DR and Obabko, AV and Yuan, Haomin and Merzari, Elia and Fischer, PF",
    title = "A robust spectral element implementation of the k- $\tau$ RANS model in Nek5000/NekRS",
    journal = "International Journal of Heat and Fluid Flow",
    volume = "112",
    pages = "109679",
    year = "2025",
    publisher = "Elsevier"
}

RIS

TY  - JOUR
AU  - Tomboulides, Ananias
AU  - Saini, Nadish
AU  - Shaver, DR
AU  - Obabko, AV
AU  - Yuan, Haomin
AU  - Merzari, Elia
AU  - Fischer, PF
TI  - A robust spectral element implementation of the k- τ RANS model in Nek5000/NekRS
JO  - International Journal of Heat and Fluid Flow
PY  - 2025
VL  - 112
PB  - Elsevier
SP  - 109679
ER  -

Plain Text

Ananias Tomboulides, Nadish Saini, DR Shaver, AV Obabko, Haomin Yuan, Elia Merzari, and PF Fischer. A robust spectral element implementation of the k- τ rans model in nek5000/nekrs. International Journal of Heat and Fluid Flow, 112:109679, 2025.

Computational domain
Flow parameters
Test Case
Verification criteria
References