Finite volume

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-	== Discretisation Schemes for convective terms in General Transport Equation. Finite-Volume Formulation, structured grids  ==	+	The Finite Volume Method (FVM) is one of the most versatile discretization techniques used in CFD. Based on the control volume formulation of analytical fluid dynamics, the first step in the FVM is to divide the domain into a number of control volumes (aka cells, elements) where the variable of interest is located at the centroid of the control volume. The next step is to integrate the differential form of the governing equations (very similar to the control volume approach) over each control volume. Interpolation profiles are then assumed in order to describe the variation of the concerned variable between cell centroids. The resulting equation is called the discretized or discretization equation. In this manner, the discretization equation expresses the conservation principle for the variable inside the control volume.
		+	The most compelling feature of the FVM is that the resulting solution satisfies the conservation of quantities such as mass, momentum, energy, and species. This is '''exactly''' satisfied for any control volume as well as for the whole computational domain and '''for any number of control volumes'''. Even a coarse grid solution exhibits exact integral balances.<br>
-	== Introduction ==	+	FVM is the ideal method for computing discontinuous solutions arising in compressible flows. Any discontinuity must satisfy the [[Rankine-Hugoniot jump condition]] which is a consequence of conservation. Since finite volume methods are conservative they automatically satisfy the jump conditions and hence give physically correct weak solutions.
-	Here is describing the discretization schemes of the convective terms in the finite-volume equations. The accuracy, numerical stability and the boundness of the solution depends on the numerical scheme used for these terms. The central issue is the specification of an appropriate relationship between the convected variable, stored at the cell centre and its value at each of the cell faces.	+	FVM is also preferred while solving partial differential equations containing discontinuous coefficients.
-	== Basic Equations of CFD ==	+	==Integral form of conservation law==
-	All the conservation equations can be written in the same generic differential form:	+	Consider a single partial differential equation in conservation form
-	<table width="100%"><tr><td>	+	:<math>
-	:<math>	+	\frac{\partial u}{\partial t} + \nabla \cdot f(u) = 0
-	\frac {\partial( \rho \phi )} {\partial t} + \frac{\partial}{\partial x_{i}} \left( \rho U \phi - \Gamma_{\phi} \frac{\partial\phi}{\partial x_{i}}\right)=S_{\phi}	+
	</math>		</math>
-	</td><td width="5%">(1)</td></tr></table>
-	Equation (1) is integrated over a control volume and the following discretised equation for <math>\boldsymbol{\phi}</math> is produced:	+	where <math>u</math> is a conserved quantity and
-	<table width="100%"><tr><td>	+	:<math>
-	:<math>	+	f(u) = [ f_1(u), \ldots, f_d(u)]
-	J_{h}- J_{l} + J_{n}- J_{s} + J_{e}- J_{w} + D_{h} - D_{l} + D_{n} - D_{s} + D_{e} - D_{s} = S_{p}	+
	</math>		</math>
-	</td><td width="5%">(2)</td></tr></table>
-	where <math>\boldsymbol{S_{p}}</math> is the source term for the control volume <math>\boldsymbol{P}</math>, and $J_{f}$ and $D_{f}$ represent, respectively, the convective and diffusive fluxes of $\phi$ across the control-volume face $f$ ($f=h,l,n,s,e,w$)	+	is the flux vector. Integrating this over any volume <math>V</math> in <math>R^d</math> we get the '''integral form''' of the conservation law
-		+
-	The convective fluxes through the cell faces are calculated as:	+
-	<table width="100%"><tr><td>	+	:<math>
-	:<math>	+	\frac{\partial}{\partial t} \int_V u dx + \oint_{\partial V} f_i n_i ds = 0
-	J_{f}=C_{f}\phi_{f}	+
	</math>		</math>
-	</td><td width="5%">(1)</td></tr></table>
-
-	where $C_{f}$ is the mass flow rate across the cell face $f$. The convected variable $\phi_{f}$ associated with this mass flow rate is usually stored at the cell centres, and thus some form of interpolation assumption must be made in order to determine its value at each cell face. The interpolation procedure employed for this operation is the subject of the various schemes proposed in the literature and the accuracy, stability and boundedness of the solution depends on the procedure used.
-
-	In general, the value of <math>\boldsymbol{\phi_{f}}</math> can be explicity formulated in terms of its neighbouring nodal values by a functional relationship of the form:
-	<table width="100%"><tr><td>	+	where the [[Greens theorem | divergence theorem]] has been used to convert the divergence term to a surface integral over the surface <math>\partial V</math> and <math>(n_1, \ldots, n_d)</math> is the unit outward normal to <math>\partial V</math>. The above equation is the starting point for the finite volume method.
-	:<math>	+
-	\phi_{f}=P  \left( \phi_{nb} \right)	+
-	</math>	+
-	</td><td width="5%">(1)</td></tr></table>	+
-		+
-	where <math>\boldsymbol{\phi_{nb}}</math> denotes the neighbouring-node <math>\boldsymbol{\phi}</math>values.	+
-	Combining equations (\ref{eq3}) through (\ref{eq4a}), the discretised equation becomes:	+
-	<table width="100%"><tr><td>	+	==Finite volume formulation==
-	:<math>	+
-	\left\{ D_{h} + C_{h} \left[ P \left( \phi_{nb} \right) \right]_{h} \right\} -	+
-	\left\{ D_{l} + C_{l} \left[ P \left( \phi_{nb} \right) \right]_{l} \right\} +	+
-	\left\{ D_{n} + C_{n} \left[ P \left( \phi_{nb} \right) \right]_{n} \right\} -	+	The computational domain <math>V_h</math> is divided into non-overlapping cells or finite volumes
-	\left\{ D_{s} + C_{s} \left[ P \left( \phi_{nb} \right) \right]_{s} \right\} +	+
-	\left\{ D_{e} + C_{e} \left[ P \left( \phi_{nb} \right) \right]_{e} \right\} -	+	:<math>V_r, r=1,\ldots, N, \quad V_h = \cup_r V_r
-	\left\{ D_{w} + C_{w} \left[ P \left( \phi_{nb} \right) \right]_{w} \right\} = S_{p}	+
	</math>		</math>
-	</td><td width="5%">(1)</td></tr></table>
-	== Convection Schemes ==	+	Usually these cells are polygons (triangles, quadrilaterals) in 2-D and polyhedra (tetrahedron, hexahedron, prims, etc) in 3-D.
-	All the convection schemes involve a stencil of cells in which the values of $\phi$ will be used to construct the face value $\phi_{f}$	+	Introduce the cell average value over the cell <math>V_r</math>
		+	:<math>
		+	u_r = \frac{1}{|V_r|} \int_{V_r} u(x) dx
		+	</math>
-		+	which is the basic unknown quantity in the finite volume method. Let
-	\begin{figure}[h]	+
-	\centering	+
-	\includegraphics{D:/TeXWorkPlace/Wikki_Article/Picture_01.jpg}	+
-	\caption{Stensil of cells}	+
-		+
-	\label{fig:Picture_01}	+
-	\end{figure}	+
-	Where flow is from left to right, and $f$ is the face in question.	+	:<math>
		+	N(r) = \{ \mbox{Set of cells which share a common face with } V_r \}
		+	</math>
		+	Then the integral conservation law for cell <math>V_r</math> is
-	== Basic Discretisation schemes ==	+	:<math>
		+	|V_r| \frac{du_r}{dt} + \sum_{s \in N(r)} \int_{V_r \cap V_s} f_i n_i ds = 0
		+	</math>
-	== Central Differencing Scheme (CDS)==	+	It now remains to approximate the flux integral. This can be achieved using gaussian quadrature. Taking <math>p</math> gaussian points we get
-	The most natural assumption for the cell-face value of the convected variable $\phi_{f}$ would appear to be the CDS, which calculates the cell-face value from:	+	:<math>
-		+	\int_{V_r \cap V_s} f_i n_i ds = \Delta s_{rs} \sum_{m=1}^p \omega_m F^m_{rs}
-	<table width="100%"><tr><td>	+
-	:<math>	+
-	\phi_{f}=0.5 \left( \phi_{c} + \phi_{d} \right)	+
	</math>		</math>
-	</td><td width="5%">(1)</td></tr></table>
-
-	This scheme is 2nd-order accurate, but is unbounded so that unphysical oscillations appear in regions of strong convection and also in the presence of discontinuities such as shocks. The CDS may be used directly in very low Reynolds-number flows where diffusive effects dominate over convection.
-	== Upwind Differencing Scheme (UDS)==	+	where <math>\omega_m</math> are the gaussian weights and <math>F</math> is an approximation to <math>f_i n_i</math>.
-	The UDS assumes that the convected variable at the cell fase $f$ is the same as the upwind cell-centre value:	+	== External links ==
		+	* [http://www.imtek.uni-freiburg.de/simulation/mathematica/imsReferencePointers/FVM_introDocu.html The Finite Volume Method (FVM) - An introduction] by Oliver Rübenkönig of Albert Ludwigs University of Freiburg, available under the GNU Free Document License|GFDL.
-	<table width="100%"><tr><td>	+
-	:<math>	+	<i> Return to [[Numerical methods | Numerical Methods]] </i>
-	\phi_{f}=  \phi_{c}	+
-	</math>	+	{{stub}}
-	</td><td width="5%">(1)</td></tr></table>	+
-		+
-	The UDS is unconditionally bounded and highly stable, but as noted earlier it is only 1st-order accurate in terms of truncation error and may produce severe numerical diffusion. The scheme is therefore highly diffusive when the flow direction is skewed relative to the grid lines.	+

---

Latest revision as of 09:15, 3 January 2012

The Finite Volume Method (FVM) is one of the most versatile discretization techniques used in CFD. Based on the control volume formulation of analytical fluid dynamics, the first step in the FVM is to divide the domain into a number of control volumes (aka cells, elements) where the variable of interest is located at the centroid of the control volume. The next step is to integrate the differential form of the governing equations (very similar to the control volume approach) over each control volume. Interpolation profiles are then assumed in order to describe the variation of the concerned variable between cell centroids. The resulting equation is called the discretized or discretization equation. In this manner, the discretization equation expresses the conservation principle for the variable inside the control volume.

The most compelling feature of the FVM is that the resulting solution satisfies the conservation of quantities such as mass, momentum, energy, and species. This is exactly satisfied for any control volume as well as for the whole computational domain and for any number of control volumes. Even a coarse grid solution exhibits exact integral balances.

FVM is the ideal method for computing discontinuous solutions arising in compressible flows. Any discontinuity must satisfy the Rankine-Hugoniot jump condition which is a consequence of conservation. Since finite volume methods are conservative they automatically satisfy the jump conditions and hence give physically correct weak solutions.

FVM is also preferred while solving partial differential equations containing discontinuous coefficients.

Integral form of conservation law

Consider a single partial differential equation in conservation form

where is a conserved quantity and

is the flux vector. Integrating this over any volume in we get the integral form of the conservation law

where the divergence theorem has been used to convert the divergence term to a surface integral over the surface and is the unit outward normal to . The above equation is the starting point for the finite volume method.

Finite volume formulation

The computational domain is divided into non-overlapping cells or finite volumes

Usually these cells are polygons (triangles, quadrilaterals) in 2-D and polyhedra (tetrahedron, hexahedron, prims, etc) in 3-D.

Introduce the cell average value over the cell

which is the basic unknown quantity in the finite volume method. Let

Then the integral conservation law for cell is

It now remains to approximate the flux integral. This can be achieved using gaussian quadrature. Taking gaussian points we get

where are the gaussian weights and is an approximation to .

External links

• The Finite Volume Method (FVM) - An introduction by Oliver Rübenkönig of Albert Ludwigs University of Freiburg, available under the GNU Free Document License|GFDL.

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