QUICK scheme

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.^[1]

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 where as QUICK does take the second order derivative into account, but ignores the third order derivative hence this is considered third order accurate.^[2] 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.^[3]

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.^[4] 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.

Φ_w when u_w > 0 and u_e > 0 a quadratic fit through WW, W and P is used,
Φ_e when u_w > 0 and u_e > 0 a quadratic fit through W, P and E is used,
Φ_w when u_w < 0 and u_e < 0 values of W, P and E are used,
Φ_e when u_w < 0 and u_e < 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:

\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(\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(\rho u) \over dx}=0.

Discretizing the above equation to a control volume around a particular node we get

(\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

\left(\rho uA\right)_{e}-\left(\rho uA\right)_{w}=0

now assuming $F=\rho u$ and $D=r/\sigma x$

The corresponding cell face values of the above variables are given by

F_{w}=\left(\rho u\right)_{w}

F_{e}=\left(\rho u\right)_{e}

D_{w}={r_{w}}/{\sigma x_{WP}}

D_{e}={r_{e}}/{\sigma x_{PE}}

Assuming constant area over the entire control volume we get

F_{e}\phi _{e}-F_{w}\phi _{w}=D_{e}\left(\phi _{E}-\phi _{P}\right)-D_{w}\left(\phi _{P}-\phi _{W}\right)

Positive direction

When the flow is in positive direction the values of the velocities will be $u_{w}>0$ and $u_{e}>0$ ,

For "w (west face)" bracketing nodes are W and P, the upstream node is WW then,^[5]

\phi _{w}={\frac {6}{8}}\phi _{W}+{\frac {3}{8}}\phi _{P}-{\frac {1}{8}}\phi _{WW}

For "e (east face)" bracketing nodes are P and E, the upstream node is W then

\phi _{e}={\frac {6}{8}}\phi _{P}+{\frac {3}{8}}\phi _{E}-{\frac {1}{8}}\phi _{W}

Gradient of parabola is used to evaluate diffusion terms.

If F_w > 0 and F_e > 0 and if we use above equations for the convective terms and central differencing for the diffusion terms, the discretized form of the one-dimensional convection–diffusion transport equation will be written as:

F_{e}\phi _{e}-F_{w}\phi _{w}=D_{e}\left(\phi _{E}-\phi _{P}\right)-D_{w}\left(\phi _{P}-\phi _{W}\right)

F_{e}\left({\frac {6}{8}}\phi _{p}+{\frac {3}{8}}\phi _{E}-{\frac {1}{8}}\phi _{w}\right)-F_{W}\left({\frac {6}{8}}\phi _{w}+{\frac {3}{8}}\phi _{p}-{\frac {1}{8}}\phi _{ww}\right)=D_{e}\left(\phi _{E}-\phi _{P}\right)-D_{W}\left(\phi _{p}-\phi _{w}\right)

On re-arranging we get

\left(D_{w}-{\frac {3}{8}}F_{w}+D_{e}+{\frac {6}{8}}F_{e}\right)\phi _{P}=\left(D_{w}+{\frac {6}{8}}F_{w}+{\frac {1}{8}}F_{e}\right)\phi _{W}+\left(D_{e}-{\frac {3}{8}}F_{e}\right)\phi _{E}-{\frac {1}{8}}F_{w}\phi _{WW}

now it can be written in the standard form:

a_{P}\phi _{P}=a_{W}\phi _{W}+a_{E}\phi _{E}+a_{WW}\phi _{WW}

where:

$a_{W}$	$a_{E}$	$a_{WW}$	$a_{P}$
$D_{w}+{\frac {6}{8}}F_{w}+{\frac {1}{8}}F_{e}$	$D_{e}-{\frac {3}{8}}F_{e}$	$-{\frac {1}{8}}F_{w}$	$a_{W}+a_{E}+a_{WW}+\left(F_{e}-F_{w}\right)$

Negative direction

When the flow is in negative direction the value of the velocities will be u_w < 0 and u_e < 0,

For west face w the bracketing nodes are W and P, upstream node is E and for the east face E the bracketing nodes are P and E, upstream node is EE

For $F_{w}$ < 0 and $F_{e}$ < 0 the flux across the west and east boundaries is given by the expressions :

\phi _{w}={\frac {6}{8}}\phi _{P}+{\frac {3}{8}}\phi _{W}-{\frac {1}{8}}\phi _{E}

\phi _{e}={\frac {6}{8}}\phi _{E}+{\frac {3}{8}}\phi _{P}-{\frac {1}{8}}\phi _{EE}

Substitution of these two formulae for the convective terms in the discretized convection-diffusion equation together with central differencing for the diffusion terms leads, after re-arrangement similar to positive direction as above, to the following coefficients.

$a_{W}$	$a_{E}$	$a_{EE}$	$a_{P}$
$D_{w}+{\frac {3}{8}}F_{w}$	$D_{e}-{\frac {6}{8}}F_{e}-{\frac {1}{8}}F_{w}$	${\frac {1}{8}}F_{e}$	$a_{W}+a_{E}+a_{EE}+\left(F_{e}-F_{w}\right)$

QUICK scheme for 1-D convection–diffusion problems

a_PΦ_P = a_WΦ_W + a_EΦ_E + a_WWΦ_WW + a_EEΦ_EE

Here, a_P = a_W + a_E + a_WW + a_EE + (F_e - F_w)

other coefficients

a_W	a_WW	a_E	a_EE
D_w + 6/8 α_w F_w + 1/8F_e α_e +3/8 (1 – α_w)F_w	−1/8 α_wF_w	D_e - 3/8α_e F_e -6/8(1–α_e)F_e −1/8 (1–α_w)F_w	1/8(1 – α_e)F_e

where

α_w=1 for F_w > 0 and α_e=1 for F_e > 0

α_w=0 for F_w < 0 and α_e=0 for F_e < 0.

Comparing the solutions of QUICK and upwind schemes

From the below graph we can see that the QUICK scheme is more accurate than the upwind scheme. In the QUICK scheme we face the problems of undershoot and overshoot due to which some errors occur. These overshoots and undershoots should be considered while interpreting solutions. False diffusion errors will be minimized with the QUICK scheme when compared with other schemes.

References

^ Leonard, B.P. (1979), "A stable and accurate convective modelling procedure based on quadratic upstream interpolation", Computer Methods in Applied Mechanics and Engineering, 19 (1): 59–98, Bibcode:1979CMAME..19...59L, doi:10.1016/0045-7825(79)90034-3
^ Versteeg, H. K.; Malalasekera, W. (1995), An introduction to computational fluid dynamics, pp. 125–132, ISBN 0-470-23515-2
^ Lin, Pengzhi (30 April 2008), Numerical Modeling of Water Waves: An Introduction to Engineers and Scientists, p. 145, ISBN 978-0-415-41578-1
^ Mitra, Sushanta K.; Chakraborty, Suman (20 September 2011), Microfluidics and Nanofluidics Handbook: Fabrication, Implementation, and Applications, p. 161, ISBN 978-1-4398-1671-4
^ Jakobsen, Hugo A. (23 June 2008), Chemical Reactor Modeling: Multiphase Reactive Flows, p. 1029, ISBN 978-3-540-25197-2