In computational fluid dynamics QUICK, which stands for Quadratic Upstream Interpolation for Convective Kinematics, is a higher-order differencing scheme that considers a three-point upstream weighted by quadratic interpolation for the cell face values. In computational fluid dynamics there are many solution methods for solving the steady convection–diffusion equation. Some of the used methods are the central differencing scheme, upwind scheme, hybrid scheme, power law scheme and QUICK scheme. The QUICK scheme was presented by Brian P. Leonard – together with the QUICKEST (QUICK with Estimated Streaming Terms) scheme – in a 1979 paper. In order to find the cell face value a quadratic function passing through two bracketing or surrounding nodes and one node on the upstream side must be used. In central differencing scheme and second order upwind scheme the first order derivative is included and the second order derivative is ignored. These schemes are therefore considered second order accurate whereas QUICK does take the second order derivative into account, but ignores the third order derivative hence this is considered third order accurate. This scheme is used to solve convection–diffusion equations using second order central difference for the diffusion term and for the convection term the scheme is third order accurate in space and first order accurate in time. QUICK is most appropriate for steady flow or quasi-steady highly convective elliptic flow.
Quadratic interpolation for QUICK scheme
For the one-dimensional domain shown in the figure the Φ value at a control volume face is approximated using three-point quadratic function passing through the two bracketing or surrounding nodes and one other node on upstream side. In the figure, in order to calculate the value of the property at the face, we should have three nodes i.e. two bracketing or surrounding nodes and one upstream node.
Φw when uw > 0 and ue > 0 a quadratic fit through WW, W and P is used, Φe when uw > 0 and ue > 0 a quadratic fit through W, P and E is used, Φw when uw < 0 and ue < 0 values of W, P and E are used, Φe when uw < 0 and ue < 0 values of P, E and EE are used. Let the two bracketing nodes be i and i − 1 and upstream node i – 2 then for a uniform grid the value of φ at the cell face between the three nodes is given by:
ϕ f a c e = 6 8 ϕ i − 1 + 3 8 ϕ i − 1 8 ϕ i − 2 {\displaystyle \phi _{face}={\frac {6}{8}}\phi _{i-1}+{\frac {3}{8}}\phi _{i}-{\frac {1}{8}}\phi _{i-2}}
Interpretation of the property when the flow is in different directions The steady convection and diffusion of a property 'Ƥ' in a given one-dimensional flow field with velocity 'u' and in the absence of sources is given
d ( ρ u ϕ ) d x = d d x ( r d ϕ d x ) . {\displaystyle {d(\rho u\phi ) \over dx}={\frac {d}{dx}}\left(r{\frac {d\phi }{dx}}\right).}
For the continuity of the flow it must also satisfy
d ( ρ u ) d x = 0. {\displaystyle {d(\rho u) \over dx}=0.}
Discretizing the above equation to a control volume around a particular node we get
( ρ u A ϕ ) e − ( ρ u A ϕ ) w = ( r A ∂ ϕ ∂ x ) e − ( r A ∂ ϕ ∂ x ) w {\displaystyle (\rho uA\phi )_{e}-(\rho uA\phi )_{w}=\left(rA{\frac {\partial \phi }{\partial x}}\right)_{e}-\left(rA{\frac {\partial \phi }{\partial x}}\right)_{w}}
Integrating this continuity equation over the control volume we get
( ρ u A ) e − ( ρ u A ) w = 0 {\displaystyle \left(\rho uA\right)_{e}-\left(\rho uA\right)_{w}=0}
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