Turbulent Channel Flow (RANS)

This case verifies the standard $var element = document.getElementById("moose-equation-7c3a0141-91e0-48ab-8494-c92de333e5f5");katex.render("k", element, {displayMode:false,throwOnError:false});$ - $var element = document.getElementById("moose-equation-019fed54-a32f-45b3-b3ff-e078db3563b1");katex.render("\\tau", element, {displayMode:false,throwOnError:false});$ Reynolds-averaged Navier–Stokes (RANS) turbulence model (Tomboulides et al., 2025) using fully developed turbulent flow in an infinite half-channel. Two simulations verify the nondimensional and dimensional formulations of the same flow.

Flow parameters

The bulk Reynolds number is $var element = document.getElementById("moose-equation-6dfe6ea3-7bb6-4da1-825c-ddd5e52c0161");katex.render("Re = 43500", element, {displayMode:false,throwOnError:false});$ and is defined as

(1)var element = document.getElementById("moose-equation-ee6feebc-0d2c-46b3-95a9-bfe648300844");katex.render("Re = \\frac{\\rho_0 U_b L}{\\mu_0},", element, {displayMode:true,throwOnError:false});

where $var element = document.getElementById("moose-equation-a5a08b2c-13a7-44a7-82a5-eb8af26f6b8b");katex.render("L", element, {displayMode:false,throwOnError:false});$ , $var element = document.getElementById("moose-equation-f0ee5681-b3f6-44e8-99e3-fcd5eacf2f51");katex.render("U_b", element, {displayMode:false,throwOnError:false});$ , $var element = document.getElementById("moose-equation-1fbae773-823f-44bc-98b7-c65d5c8c9caa");katex.render("\\rho_0", element, {displayMode:false,throwOnError:false});$ , and $var element = document.getElementById("moose-equation-d6fb0e52-46b2-4b8e-b9cc-d7861c54fd3a");katex.render("\\mu_0", element, {displayMode:false,throwOnError:false});$ are the dimensional reference length, bulk velocity, density, and dynamic viscosity, respectively. The corresponding friction Reynolds number, based on the friction velocity $var element = document.getElementById("moose-equation-ac002aa7-2da3-4d25-b1ee-454e2cc11a7f");katex.render("u_\\tau", element, {displayMode:false,throwOnError:false});$ , is approximately 2000 and is defined as

(2)var element = document.getElementById("moose-equation-09147c76-478c-413a-a98e-80569f6bc0fc");katex.render("Re_\\tau = \\frac{\\rho_0 u_\\tau L}{\\mu_0},", element, {displayMode:true,throwOnError:false});

where

(3)var element = document.getElementById("moose-equation-e003c4c3-7dc3-41bf-81e6-d2e2b0b54a33");katex.render("u_\\tau = \\sqrt{\\frac{\\tau_w}{\\rho_0}}.", element, {displayMode:true,throwOnError:false});

Here, $var element = document.getElementById("moose-equation-c91f3c43-a39e-4f1e-b426-fb6bc4029cd5");katex.render("\\tau_w", element, {displayMode:false,throwOnError:false});$ is the computed wall shear stress.

Nondimensional formulation

The first test case uses the nondimensional formulation with

(4)var element = document.getElementById("moose-equation-cfcd891d-0e0c-4b76-8e4a-bea6987e42e6");katex.render("L^* = 1, \\qquad U_b^* = 1, \\qquad \\rho^* = 1, \\qquad \\mu^* = \\frac{1}{Re},", element, {displayMode:true,throwOnError:false});

where the superscript $var element = document.getElementById("moose-equation-bc6d2b81-4056-4484-97d1-ed0b750ace95");katex.render("*", element, {displayMode:false,throwOnError:false});$ denotes a nondimensional quantity. The computed nondimensional friction velocity is compared with the Direct Numerical Simulation (DNS) reference of Lee et al.* (Lee and Moser, 2015),

(5)var element = document.getElementById("moose-equation-0076416f-b68a-40a3-9054-ff87692d6809");katex.render("u_{\\tau,\\mathrm{DNS}}^* = 4.58794 \\times 10^{-2}.", element, {displayMode:true,throwOnError:false});

Dimensional formulation

Th second test case represents the same flow using the dimensional reference scales $var element = document.getElementById("moose-equation-d9251771-d943-4d83-a4b7-523a1f527af5");katex.render("L", element, {displayMode:false,throwOnError:false});$ , $var element = document.getElementById("moose-equation-48d4d480-142b-4576-bbf7-c9d7ba65b0cf");katex.render("U_b", element, {displayMode:false,throwOnError:false});$ , $var element = document.getElementById("moose-equation-75d71576-021c-4f9b-81cb-5747e7833d3f");katex.render("\\rho_0", element, {displayMode:false,throwOnError:false});$ , and $var element = document.getElementById("moose-equation-0382b665-cf21-4992-95c9-2313dc40e5a7");katex.render("\\mu_0", element, {displayMode:false,throwOnError:false});$ . The dimensional dynamic viscosity is selected to preserve the bulk Reynolds number,

(6)var element = document.getElementById("moose-equation-27c2aae8-6c3e-4a8a-9dd2-1aec99f15503");katex.render("\\mu_0 = \\frac{\\rho_0 U_b L}{Re}.", element, {displayMode:true,throwOnError:false});

The corresponding dimensional DNS friction velocity is

(7)var element = document.getElementById("moose-equation-f9ee3a17-c880-4e9c-a71f-9e3416ba01a0");katex.render("u_{\\tau,\\mathrm{DNS}} = U_b u_{\\tau,\\mathrm{DNS}}^*.", element, {displayMode:true,throwOnError:false});

Verification criterion

For both simulations, the relative error is evaluated using

(8)var element = document.getElementById("moose-equation-336dae6a-3432-48d1-a7aa-43a9a298fccc");katex.render("\\mathrm{err} = \\frac{\\left|u_\\tau-u_{\\tau,\\mathrm{DNS}}\\right|}{u_{\\tau,\\mathrm{DNS}}}.", element, {displayMode:true,throwOnError:false});

The nondimensional quantities $var element = document.getElementById("moose-equation-33a839f4-3e77-48fc-bc6c-471020ce55f1");katex.render("u_\\tau^*", element, {displayMode:false,throwOnError:false});$ and $var element = document.getElementById("moose-equation-b64b5ceb-466f-4461-aed6-34a8e10e9f41");katex.render("u_{\\tau,\\mathrm{DNS}}^*", element, {displayMode:false,throwOnError:false});$ are used when the solve is performed in non-dimensional form. This comparison verifies that dimensional scaling preserves the predicted friction velocity and the nondimensional flow behavior.

References

Myoungkyu Lee and Robert D Moser. Direct numerical simulation of turbulent channel flow up to. Journal of fluid mechanics, 774:395–415, 2015.

BibTeX

@article{lee2015direct,
    author = "Lee, Myoungkyu and Moser, Robert D",
    title = "Direct numerical simulation of turbulent channel flow up to",
    journal = "Journal of fluid mechanics",
    volume = "774",
    pages = "395--415",
    year = "2015",
    publisher = "Cambridge University Press"
}

RIS

TY  - JOUR
AU  - Lee, Myoungkyu
AU  - Moser, Robert D
TI  - Direct numerical simulation of turbulent channel flow up to
JO  - Journal of fluid mechanics
PY  - 2015
VL  - 774
PB  - Cambridge University Press
SP  - 395
EP  - 415
ER  -

Plain Text

Myoungkyu Lee and Robert D Moser. Direct numerical simulation of turbulent channel flow up to. Journal of fluid mechanics, 774:395–415, 2015.

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-1affa665-2291-4e78-980c-a5578864dc5b");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.

Flow parameters
Nondimensional formulation
Dimensional formulation
Verification criterion
References