Non-Dimensional Formulation

NekRS can solve its equations in dimensional or non-dimensional form. When solving in non-dimensional form, characteristic scales for velocity, temperature, length, and time are defined and substituted into the Navier-Stokes equations so that all solution fields (velocity, pressure, temperature) are of order unity. Non-dimensional formulations for velocity, pressure, temperature, length, and time are defined as

$(1)var element = document.getElementById("moose-equation-8018dd7f-874d-45f1-ba91-aafdcaf4815e");katex.render(" u_i^\\dagger\\equiv\\frac{u_i}{u_{ref}}", element, {displayMode:true,throwOnError:false});$

$(2)var element = document.getElementById("moose-equation-75f33096-5a7a-4850-8b53-bc09776cf8f9");katex.render(" P^\\dagger\\equiv\\frac{P}{\\rho_0u_{ref}^2}", element, {displayMode:true,throwOnError:false});$

$(3)var element = document.getElementById("moose-equation-ca9449e0-eb59-4336-83f9-d734f1086c96");katex.render(" T^\\dagger\\equiv\\frac{T-T_{ref}}{\\Delta T}", element, {displayMode:true,throwOnError:false});$

$(4)var element = document.getElementById("moose-equation-590ecd04-e3b7-4272-b518-9ebc219181ab");katex.render(" x_i^\\dagger\\equiv\\frac{x_i}{L_{ref}}", element, {displayMode:true,throwOnError:false});$

$(5)var element = document.getElementById("moose-equation-2b7c4b72-60c6-4cbf-bf8a-c59653c2dd6f");katex.render(" t^\\dagger\\equiv\\frac{t}{L_{ref}/u_{ref}}", element, {displayMode:true,throwOnError:false});$

where a $var element = document.getElementById("moose-equation-9944873a-9e4c-4b57-a636-f9710910f58e");katex.render("\\dagger", element, {displayMode:false,throwOnError:false});$ superscript indicates a non-dimensional quantity and a $var element = document.getElementById("moose-equation-6c5c153a-3a7c-440e-a73d-b47c42aea08e");katex.render("ref", element, {displayMode:false,throwOnError:false});$ subscript indicates a characteristic scale. A " $var element = document.getElementById("moose-equation-044379fe-cd83-4fd9-a1c0-f5d9b9933d2e");katex.render("0", element, {displayMode:false,throwOnError:false});$ " subscript indicates a reference parameter (so-called because a reference parameter is not really a characteristic scale, but rather a reference value corresponding to the conditions at the characteristic scales). Inserting these definitions into the conservation of mass, momentum, and energy equations solved by NekRS gives

$(6)var element = document.getElementById("moose-equation-1cdb9a90-b3e9-4365-b220-1b126aaa1dfb");katex.render(" \\frac{\\partial u_i^\\dagger}{\\partial x_i^\\dagger}=0", element, {displayMode:true,throwOnError:false});$

$(7)var element = document.getElementById("moose-equation-da19f486-c357-48da-afea-b6808a953e3e");katex.render(" \\rho^\\dagger\\left(\\frac{\\partial u_i^\\dagger}{\\partial t^\\dagger}+u_j^\\dagger\\frac{\\partial u_i^\\dagger}{\\partial x_j^\\dagger}\\right)=-\\frac{\\partial P^\\dagger}{\\partial x_i^\\dagger}+\\frac{1}{Re}\\frac{\\partial \\tau_{ij}^\\dagger}{\\partial x_j^\\dagger}+\\rho^\\dagger f_i^\\dagger", element, {displayMode:true,throwOnError:false});$

$(8)var element = document.getElementById("moose-equation-c18dea98-4f76-4d76-b15c-6da61fa919c7");katex.render(" \\rho^\\dagger C_p^\\dagger\\left(\\frac{\\partial T^\\dagger}{\\partial t^\\dagger}+u_i^\\dagger\\frac{\\partial T^\\dagger}{\\partial x_i^\\dagger}\\right)=\\frac{1}{Pe}\\frac{\\partial}{\\partial x_i^\\dagger}\\left(k^\\dagger\\frac{\\partial T^\\dagger}{\\partial x_i^\\dagger}\\right)+\\dot{q}^\\dagger", element, {displayMode:true,throwOnError:false});$

New terms in these non-dimensional equations are $var element = document.getElementById("moose-equation-331e331b-9db6-4f99-9107-6a139842654b");katex.render("Re", element, {displayMode:false,throwOnError:false});$ and $var element = document.getElementById("moose-equation-b0000202-da32-4a96-9ba1-a572a1ae208f");katex.render("Pe", element, {displayMode:false,throwOnError:false});$ , the Reynolds and Peclet numbers, respectively:

$(9)var element = document.getElementById("moose-equation-b570e86b-7ccd-4d1e-85a8-73b6db1bf8c6");katex.render(" Re\\equiv\\frac{\\rho_0 u_{ref}L_{ref}}{\\mu_0}", element, {displayMode:true,throwOnError:false});$

$(10)var element = document.getElementById("moose-equation-24a38c03-b0fd-4f34-b761-364a5b9170c0");katex.render(" Pe\\equiv\\frac{L_{ref}u_{ref}}{\\alpha}", element, {displayMode:true,throwOnError:false});$

where $var element = document.getElementById("moose-equation-c1cfbc60-5221-455b-94e0-4aaf2cb5eeb8");katex.render("\\alpha", element, {displayMode:false,throwOnError:false});$ is the thermal diffusivity. NekRS solves for $var element = document.getElementById("moose-equation-2b40dea2-9e36-43d2-980b-d6f29a43655f");katex.render("\\mathbf u^\\dagger", element, {displayMode:false,throwOnError:false});$ , $var element = document.getElementById("moose-equation-78c59e61-37d3-4eaf-8747-316a8bf78c12");katex.render("P^\\dagger", element, {displayMode:false,throwOnError:false});$ , and $var element = document.getElementById("moose-equation-79709af3-7fac-4615-96a0-2ba9d10c7996");katex.render("T^\\dagger", element, {displayMode:false,throwOnError:false});$ . Cardinal will handle conversions from a non-dimensional NekRS solution to a dimensional MOOSE application.