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

$(1)var element = document.getElementById("moose-equation-ded71b98-878a-4c89-b538-51c676121809");katex.render(" u_i^\\dagger\\equiv\\frac{u_i}{u_{ref}}", element, {displayMode:true,throwOnError:false});$

$(2)var element = document.getElementById("moose-equation-350a22c5-201e-42d3-9961-69ead38df31e");katex.render(" P^\\dagger\\equiv\\frac{P}{\\rho_0u_{ref}^2}", element, {displayMode:true,throwOnError:false});$

$(3)var element = document.getElementById("moose-equation-1f787cae-f90a-4e3f-a64f-a0115df2776f");katex.render(" T^\\dagger\\equiv\\frac{T-T_{ref}}{\\Delta T}", element, {displayMode:true,throwOnError:false});$

$(4)var element = document.getElementById("moose-equation-105d7722-d573-40e2-82fc-abc90c743d8d");katex.render(" x_i^\\dagger\\equiv\\frac{x_i}{L_{ref}}", element, {displayMode:true,throwOnError:false});$

$(5)var element = document.getElementById("moose-equation-98878716-0bda-4d14-a709-3f490688e4c6");katex.render(" t^\\dagger\\equiv\\frac{t}{L_{ref}/u_{ref}}", element, {displayMode:true,throwOnError:false});$

where a $var element = document.getElementById("moose-equation-e73512bd-ba94-4224-8cf5-4ce68944733d");katex.render("\\dagger", element, {displayMode:false,throwOnError:false});$ superscript indicates a non-dimensional quantity and a $var element = document.getElementById("moose-equation-4b331260-ad88-4b63-88b0-af2a2b0951f9");katex.render("ref", element, {displayMode:false,throwOnError:false});$ subscript indicates a characteristic scale. A " $var element = document.getElementById("moose-equation-5aa385e6-766c-4b85-bb9e-4d65cef30d28");katex.render("0", element, {displayMode:false,throwOnError:false});$ " 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

$(6)var element = document.getElementById("moose-equation-d730be38-bf93-4df1-a47b-8b7cdb95271c");katex.render(" \\frac{\\partial u_i^\\dagger}{\\partial x_i^\\dagger}=0", element, {displayMode:true,throwOnError:false});$

$(7)var element = document.getElementById("moose-equation-bb5b082e-d48b-4fac-a2f9-74dc40cf1d81");katex.render(" \\rho^\\dagger\\left(\\frac{\\partial u_i^\\dagger}{\\partial t^\\dagger}+u_j^\\dagger\\frac{\\partial u_i^\\dagger}{\\partial x_j^\\dagger}\\right)=-\\frac{\\partial P^\\dagger}{\\partial x_i^\\dagger}+\\frac{1}{Re}\\frac{\\partial \\tau_{ij}^\\dagger}{\\partial x_j^\\dagger}+\\rho^\\dagger f_i^\\dagger", element, {displayMode:true,throwOnError:false});$

$(8)var element = document.getElementById("moose-equation-71aaf84f-0bf5-437f-a29d-cf5c74fa78d6");katex.render(" \\rho^\\dagger C_p^\\dagger\\left(\\frac{\\partial T^\\dagger}{\\partial t^\\dagger}+u_i^\\dagger\\frac{\\partial T^\\dagger}{\\partial x_i^\\dagger}\\right)=\\frac{1}{Pe}\\frac{\\partial}{\\partial x_i^\\dagger}\\left(k^\\dagger\\frac{\\partial T^\\dagger}{\\partial x_i^\\dagger}\\right)+\\dot{q}^\\dagger", element, {displayMode:true,throwOnError:false});$

New terms in these non-dimensional equations are $var element = document.getElementById("moose-equation-8c8deb68-fca2-486f-9f55-7390da5b42ac");katex.render("Re", element, {displayMode:false,throwOnError:false});$ and $var element = document.getElementById("moose-equation-927fa7c5-adc5-40cf-8278-51a6320ae7ed");katex.render("Pe", element, {displayMode:false,throwOnError:false});$ , the Reynolds and Peclet numbers, respectively:

$(9)var element = document.getElementById("moose-equation-1d018dc0-984f-4eb3-9a5b-97d6825c3625");katex.render(" Re\\equiv\\frac{\\rho_0 u_{ref}L_{ref}}{\\mu_0}", element, {displayMode:true,throwOnError:false});$

$(10)var element = document.getElementById("moose-equation-b9f27c5f-27d5-4625-bbe3-a0d5c0c39555");katex.render(" Pe\\equiv\\frac{L_{ref}u_{ref}}{\\alpha}", element, {displayMode:true,throwOnError:false});$

where $var element = document.getElementById("moose-equation-694fc969-aa33-4f5c-808e-c4c2d772a63b");katex.render("\\alpha", element, {displayMode:false,throwOnError:false});$ is the thermal diffusivity. NekRS solves for $var element = document.getElementById("moose-equation-c17d4ccd-81c2-4541-b8d2-51c614ed73fb");katex.render("\\mathbf u^\\dagger", element, {displayMode:false,throwOnError:false});$ , $var element = document.getElementById("moose-equation-dab414db-ebff-4336-ac75-555a40b03401");katex.render("P^\\dagger", element, {displayMode:false,throwOnError:false});$ , and $var element = document.getElementById("moose-equation-537e990b-418e-4c33-ae62-ccdb60f8a37b");katex.render("T^\\dagger", element, {displayMode:false,throwOnError:false});$ . Cardinal will handle conversions from a non-dimensional NekRS solution to a dimensional MOOSE application.