

Line 1: 
Line 1: 
  Michail, if possible, please move the convection related things to convective descretisation, have a look at numerical method page, i made some changes.
 
  thanks
 
  Arjun
 
   
  == Discretisation Schemes for convective terms in General Transport Equation. FiniteVolume Formulation, structured grids ==
 
 
 
 
 
  == Introduction ==
 
 
 
  Here is describing the discretization schemes of the convective terms in the finitevolume 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.
 
 
 
  == Basic Equations of CFD ==
 
 
 
  All the conservation equations can be written in the same generic differential form:
 
 
 
  <table width="100%"><tr><td>
 
  :<math>
 
  \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>
 
  </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:
 
 
 
  <table width="100%"><tr><td>
 
  :<math>
 
  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>
 
  </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 <math>\boldsymbol{J_{f}}</math> and <math>\boldsymbol{D_{f}}</math> represent, respectively, the convective and diffusive fluxes of <math>\boldsymbol{\phi}</math> across the controlvolume face <math>\boldsymbol{f}</math>
 
  <math>\boldsymbol{(f=h,l,n,s,e,w)}</math>
 
 
 
  The convective fluxes through the cell faces are calculated as:
 
 
 
  <table width="100%"><tr><td>
 
  :<math>
 
  J_{f}=C_{f}\phi_{f}
 
  </math>
 
  </td><td width="5%">(1)</td></tr></table>
 
 
 
  where <math>C_{f}</math> is the mass flow rate across the cell face <math>f</math>. The convected variable <math>\phi_{f}</math> 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>
 
  :<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 neighbouringnode <math>\boldsymbol{\phi}</math>values.
 
  Combining equations (\ref{eq3}) through (\ref{eq4a}), the discretised equation becomes:
 
 
 
  <table width="100%"><tr><td>
 
  :<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\} 
 
  \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\} 
 
  \left\{ D_{w} + C_{w} \left[ P \left( \phi_{nb} \right) \right]_{w} \right\} = S_{p}
 
  </math>
 
  </td><td width="5%">(1)</td></tr></table>
 
 
 
  == Convection Schemes ==
 
 
 
  All the convection schemes involve a stencil of cells in which the values of <math>\boldsymbol{\phi}</math> will be used to construct the face value <math>\boldsymbol{\phi_{f}}</math>
 
 
 
  [[Image:picture_01.jpg]].
 
 
 
 
 
  Where flow is from left to right, and <math>\boldsymbol{f}</math> is the face in question.
 
 
 
  <math>\boldsymbol{u}</math>  mean Upstream node
 
 
 
  <math>\boldsymbol{c}</math>  mean Central node
 
 
 
  <math>\boldsymbol{d}</math>  mean Downstream node
 
 
 
  == Basic Discretisation schemes ==
 
 
 
  === Central Differencing Scheme (CDS)===
 
 
 
  The most natural assumption for the cellface value of the convected variable <math>\boldsymbol{\phi_{f}}</math> would appear to be the CDS, which calculates the cellface value from:
 
 
 
  <table width="100%"><tr><td>
 
  :<math>
 
  \phi_{f}=0.5 \left( \phi_{c} + \phi_{d} \right)
 
  </math>
 
  </td><td width="5%">(1)</td></tr></table>
 
 
 
  This scheme is 2ndorder 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 Reynoldsnumber flows where diffusive effects dominate over convection.
 
 
 
  === Upwind Differencing Scheme (UDS)===
 
 
 
  The UDS assumes that the convected variable at the cell fase <math>\boldsymbol{f}</math> is the same as the upwind cellcentre value:
 
 
 
  <table width="100%"><tr><td>
 
  :<math>
 
  \boldsymbol{\phi_{f}= \phi_{c} }
 
  </math>
 
  </td><td width="5%">(1)</td></tr></table>
 
 
 
  The UDS is unconditionally bounded and highly stable, but as noted earlier it is only 1storder 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.
 
 
 
  === Hybrid Differencing Scheme (HDS) ===
 
 
 
  The HDS of Spalding [1972] switches the discretisation of the convection terms between CDS and UDS according to the local cell Peclet number as follows:
 
 
 
  <table width="100%"><tr><td>
 
  :<math>
 
  \phi_{f}=0.5 \left( \phi_{c} + \phi_{d} \right) for Pe \triangleleft 2
 
 
 
  </math>
 
  </td><td width="5%">(1)</td></tr></table>
 
 
 
  <table width="100%"><tr><td>
 
  :<math>
 
 
 
  \phi_{f}= \phi_{c} for Pe \triangleright 2
 
 
 
  </math>
 
  </td><td width="5%">(1)</td></tr></table>
 
 
 
  The cell Peclet number is defined as:
 
 
 
  <table width="100%"><tr><td>
 
  :<math>
 
  Pe= \rho \left U_{f} \right A_{f}/D_{f}
 
  </math>
 
  </td><td width="5%">(1)</td></tr></table>
 
 
 
 
 
  in which <math>\boldsymbol{A_{f}}</math> and <math>\boldsymbol{D_{f}}</math> are respectively, the cellface area and physical diffusion coefficient. When <math>\boldsymbol{Pe\triangleright 2}</math>
 
  ,CDS calculations tends to become unstable so that theHDS reverts to the UDS. Physical diffusion is ignored when <math>\boldsymbol{Pe\triangleright 2}</math>.
 
 
 
 
 
  The HDS scheme is marginally more accurate than the UDS, because the 2ndorder CDS will be used in regions of low Peclet number.
 
 
 
  == High Resolution Schemes (HRS) ==
 
 
 
  === Classification of High Resolution Schemes ===
 
 
 
  HRS can be classified as ''linear'' or ''nonlinear'', where ''linear'' means their coefficients are not direct functions of the convected variable when applied to a linear convection equation. It is important to recognise that linear convection schemes of 2ndorder accuracy or higher may suffer from unboudedness, and are not unconditionally stable.
 
 
 
  ''Nonlinear'' schemes analyse the solution within the stencil and adapt the discretisation to avoid any unwanted behavior, such as unboundedness (see Waterson [1994]). These two types of schemes may be presented in a unified way by use of the ''FluxLimiter'' formulation (Waterson and Deconinck [1995]), which calculates the cellface value of the convected variable from:
 
 
 
  <table width="100%"><tr><td>
 
  :<math>
 
  \phi_{f}= \phi_{c} + 0.5 B \left( r \right) \left( \phi_{c}\phi_{u} \right)
 
  </math>
 
  </td><td width="5%">(1)</td></tr></table>
 
 
 
 
 
  where <math>\boldsymbol{B \left( r \right)}</math> is termed a limiter function and the gradient ration <math>\boldsymbol{r}</math> is defined as:
 
 
 
 
 
  <table width="100%"><tr><td>
 
  :<math>
 
  r= \left( \phi_{d}  \phi_{c} \right) / \left( \phi_{c}  \phi_{u} \right)
 
  </math>
 
  </td><td width="5%">(1)</td></tr></table>
 
 
 
  The generalisation of this approach to handle nonuniform meshes has been given by Waterson [1994]
 
 
 
  From equation (\ref{eq9}) it can be seen that <math>\boldsymbol{B=1}</math> gives the UDS and <math>\boldsymbol{B=r}</math> gives the CDS.
 
 
 
 
 
 
 
  '''N.P.Waterson and H.Deconinck''',A unified approach to the desing and application of bounded highorder covection schemes, VKI preprint 199521, (1995)
 
 
 
  '''N.P.Waterson''', Development of bounded highorder convection scheme for general industrial applications, VKI Project Report 199433, (1994)
 
 
 
  === Numerical Implementation of HRS ===
 
 
 
  The HRS schemes can be introdused into equation (\ref{eq4b}) by using the deffered correction procedure of Rubin and Khosla [1982]. This procedure express the cellface value <math>\boldsymbol{\phi_{f}}</math> by:
 
 
 
 
 
  <table width="100%"><tr><td>
 
  :<math>
 
  \phi_{f}=\phi_{f}\left(U \right) + \phi^{'}_{f}
 
  </math>
 
  </td><td width="5%">(1)</td></tr></table>
 
 
 
  where <math>\boldsymbol{\phi^{'}_{f}}</math> is a higherorder correction which represents the difference between the UDS face value <math>\boldsymbol{\phi_{f}\left(U \right)}</math> and the higherorder scheme value <math>\boldsymbol{\phi_{f}\left(H \right)}</math> , i.e.
 
 
 
  <table width="100%"><tr><td>
 
  :<math>
 
  \phi^{'}_{f}= \phi_{f}\left(H \right) + \phi_{f}\left(U \right)
 
  </math>
 
  </td><td width="5%">(1)</td></tr></table>
 
 
 
  If equation (\ref{eq10a}) is substituted into equation (\ref{eq4b}), the resulting discretised equation is:
 
 
 
  <table width="100%"><tr><td>
 
  :<math>
 
  \left\{ D_{h} + C_{h} \phi_{h} \left( U \right) \right\} 
 
  \left\{ D_{l} + C_{l} \phi_{l} \left( U \right) \right\} +
 
 
 
  \left\{ D_{n} + C_{n} \phi_{n} \left( U \right) \right\} 
 
  \left\{ D_{s} + C_{s} \phi_{s} \left( U \right) \right\} +
 
 
 
  \left\{ D_{e} + C_{e} \phi_{e} \left( U \right) \right\} 
 
  \left\{ D_{w} + C_{w} \phi_{w} \left( U \right) \right\} = S_{p} + B_{p}
 
  </math>
 
  </td><td width="5%">(1)</td></tr></table>
 
 
 
 
 
  where <math>\boldsymbol{B_{p}}</math> is the deferredcorrection source terms, given by:
 
 
 
  <table width="100%"><tr><td>
 
  :<math>
 
  B_{p} = C_{l}\phi^{'}_{l}  C_{h}\phi^{'}_{h} +
 
  C_{s}\phi^{'}_{s}  C_{n}\phi^{'}_{n} +
 
  C_{w}\phi^{'}_{w}  C_{e}\phi^{'}_{e}
 
  </math>
 
  </td><td width="5%">(1)</td></tr></table>
 
 
 
  This treatment leads to a diagonally dominant coefficient matrix since it is formed using the UDS.
 
 
 
  The final form of the discretised equation:
 
 
 
  <table width="100%"><tr><td>
 
  :<math>
 
  a_{P}\phi_{P}= a_{N}\phi_{N} + a_{S}\phi_{S} + a_{E}\phi_{E} + a_{W}\phi_{W} + a_{H}\phi_{H} + a_{L}\phi_{L} + a_{T}\phi_{T} + S_{p} + B_{p}
 
  </math>
 
  </td><td width="5%">(1)</td></tr></table>
 
 
 
 
 
  Subscrit <math>\boldsymbol{P}</math> represents the current computational cell; <math>\boldsymbol{N, S, E, W, H, L}</math>
 
  represent the six neighbouring cells and <math>\boldsymbol{T}</math> represents the previous timestep (transistent cases only)
 
 
 
  The coefficients contain the appropriate contributions from the transient, convective and diffusive terms in (\ref{eq1})
 
 
 
 
 
 
 
  '''S.G.Rubin and P.K.Khoshla''',Polinomial interpolation method for viscous flow calculations, J. Comp. Phys., Vol. 27, p153, (1982)
 
 
 
  == Normalised Variables ==
 
 
 
  == Normalised Variables Diagram (NVD) ==
 
 
 
  == Total Variation Diminishing (TVD) ==
 
 
 
  == Convection Boundedness Criterion (CBC) ==
 
 
 
 
 
  '''Choi S.K.,Nam H.Y.,Cho M.''' A comparison of highorder bounded convection schemes // Computational Methods in Applied Mechanics and engineering. 1995 121. 281301
 
 
 
  '''Gaskell P.H., Lau A.K.C.''' Curvativecompemsated convective transport: SMART, a new boundednesspreserving trasport algorithm // Internatioan Journal for Numerical Methods in Fluids. 1988. 8, N 6. 617641
 
 
 
  == Schemes ==
 
 
 
  === QUICK  Quadratic Upwind Interpolation for Convective Kinematics ===
 
 
 
  === LUS ===
 
 
 
  === Fromm  Fromm's Upwind Scheme ===
 
 
 
  === CUS  Cubic Upwind Difference Scheme ===
 
 
 
  === van Leer limiter ===
 
 
 
  === ChakravarthyOsher limiter ===
 
 
 
  === Sweby \Phi  limiter ===
 
 
 
  === OSPRE ===
 
 
 
  === Superbee ===
 
 
 
  === MINMOD ===
 
 
 
  === ISNAS ===
 
 
 
  === MUSCL  Monotonic Upwind Scheme for Conservation Laws ===
 
 
 
  === UMIST ===
 
 
 
  === SOUCUP  SecondOrder Upwind Central differncefirst order UPwind ===
 
 
 
 
 
  Zhu J. On the higherorder bounded discretization schemes for finite volume compuyations of incompressible flows // Computational Methods in Applied Mechanics and Engineering. 1992. 98. 345360
 
 
 
  === HLPA ===
 
 
 
  Zhu J. Low Diffusive and oscillationfree convection scheme // Communications and Applied Numerical Methods. 1991. 7, N3. 225232.
 
 
 
  Zhu J., Rodi W. A low dispersion and bounded discretization schemes for finite volume computations of incompressible flows // Computational Methods for Applied Mechanics and Engineering. 1991. 92. 8796
 
 
 
  === SMART ===
 
 
 
  === SMARTER ===
 
 
 
  === LPPA ===
 
 
 
  === SHARP ===
 
 
 
  === CHARM ===
 
 
 
  === VONOS ===
 
 
 
 
 
  === CUBISTA Convergent and Universally bounded Interpolation Scheme for the Treatment of Advection ===
 
 
 
  '''M.A. Alves, P.J.Oliveira, F.T. Pinho''', A convergent and Universally Bounded Interpolation Scheme for the Treatment of Advection // International Lournal For Numerical Methods in Fluids 2003, 41; 4775
 
 
 
  == Summary of Discretizations Schemes ==
 
 
 
  == Numerical examples ==
 
 
 
  === Pure convection of a scalar step by a rotating velocity field (SmithHutton test) ===
 
 
 
  == Example code for solving SmithHutton test ==
 