Flow Boundaries

Flow boundary components are attached to ends of flow channel components to provide boundary conditions.

Usage

A flow boundary component connects to a flow channel end using the input parameter. The value of this parameter corresponds to the boundary name corresponding to a specific end of a flow channel. The format for the input parameter is flow_channel_name:in or flow_channel_name:out, where flow_channel_name is the name of the connected flow channel, and in or out depends on which end is being connected. The in side denotes the "starting" side of the flow channel, which resides at the location specified by the flow channel's position parameter, and the out side is the opposite end.

Formulation

Flow boundary formulations differ based on how the boundary flux $var element = document.getElementById("moose-equation-8a34cd7f-8a01-406c-889f-faf0dee64beb");katex.render("\\mathbf{F}_\\text{b}", element, {displayMode:false,throwOnError:false});$ is computed.

Direct Flux Computation

In this case, the boundary flux is computed directly from the continuous flux function $var element = document.getElementById("moose-equation-4ed97aa5-1fb8-4db7-914d-b64f071e8140");katex.render("\\mathbf{f}(\\mathbf{u})", element, {displayMode:false,throwOnError:false});$ using some solution vector $var element = document.getElementById("moose-equation-d56c47e5-3b28-4522-ac09-2dbc9fd282d5");katex.render("\\mathbf{U}_\\text{b}", element, {displayMode:false,throwOnError:false});$ , which is determined by the boundary condition and is in general a combination of the interior solution and exterior boundary data: $var element = document.getElementById("moose-equation-9c0fab68-16b2-4a91-88d5-98d43bed88e4");katex.render(" \\mathbf{F}_\\text{b} = \\mathcal{f}(\\mathbf{U}_b) \\eqp", element, {displayMode:true,throwOnError:false});$

Ghost Cell Flux Computation

In this case, the boundary flux is computed using a ghost cell approach. That is, the flux $var element = document.getElementById("moose-equation-af62c9df-56a6-4158-83bc-b6390a4f10c9");katex.render("\\mathbf{F}_\\text{b}", element, {displayMode:false,throwOnError:false});$ corresponding to the boundary is computed using a numerical flux function $var element = document.getElementById("moose-equation-e2142a33-7385-4a7f-8bce-31161af3ab88");katex.render("\\mathcal{F}(\\mathbf{U}_L, \\mathbf{U}_R)", element, {displayMode:false,throwOnError:false});$ , taking the interior solution $var element = document.getElementById("moose-equation-0e94ec5f-8407-4aa1-a82f-1f0d0583c599");katex.render("\\mathbf{U}_i", element, {displayMode:false,throwOnError:false});$ and ghost cell solution $var element = document.getElementById("moose-equation-c2cd3406-f91f-400d-9036-548ce22c8b44");katex.render("\\mathbf{U}_\\text{ghost}", element, {displayMode:false,throwOnError:false});$ as inputs: $var element = document.getElementById("moose-equation-c638431f-842a-41e0-8ed6-3176124b8b18");katex.render(" \\mathbf{F}_\\text{b} = \\mathcal{F}(\\mathbf{U}_i, \\mathbf{U}_\\text{ghost}) \\eqc", element, {displayMode:true,throwOnError:false});$ in this case for the "right" boundary of the flow channel.

Each boundary condition is defined by its definition of $var element = document.getElementById("moose-equation-0e47a5e4-7dc6-460c-85ee-303a741bc20f");katex.render("\\mathbf{U}_\\text{ghost}", element, {displayMode:false,throwOnError:false});$ and in general is formed by taking some values from the interior solution and some from external data.

Usage
Formulation