5. PRESSURE AND VELOCITY SPRING Each component of momentum satisfies its own scalar-transport equation. For one cell:

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1 5. PRESSURE AND VELOCITY SPRING The momentum equation 5.2 Pressure-velocity couling 5.3 Pressure-correction methods Summary References Examles 5.1 The Momentum Equation Each comonent of momentum satisfies its own scalar-transort equation. For one cell: (1) where C is the mass flux through a cell face. For the momentum-equation comonents: concentration, velocity comonent ( = u, v, w) diffusivity, Γ viscosity, μ source non-viscous forces However, these equations differ from those for assive scalars (those not affecting the flow) because they are: non-linear; couled; required also to be mass-consistent. For examle, the x-momentum flux through an x-directed face is The mass flux C is not constant but changes with u. The momentum equation is therefore non-linear and must be solved iteratively. v u mass flux ua Similarly, the y-momentum flux through an x-directed face is The v equation deends on the solution of the u equation (and vice versa). Hence, the momentum equations are couled and must be solved together. Pressure also aears in each momentum equation. This, together with the fact that velocity comonents u, v, w must satisfy mass conservation as well as momentum, further coules the equations and requires a means of determining ressure. The means of doing so is very different in comressible and incomressible CFD. In comressible flow, continuity rovides a transort equation for density ρ. Pressure is obtained by solving an energy equation to find temerature T and then using an equation of state (e.g. = ρrt). In incomressible flow, any density variations are, by definition, not determined by ressure. A ressure equation arises from the requirement that the solutions of the CFD 5 1 David Asley

2 momentum equations are also mass-consistent. In other words, mass conservation actually leads to a ressure equation! In most incomressible-flow solvers mass and momentum equations are solved sequentially and iteratively. This is called a segregated aroach, as in the following seudocode: DO WHILE (not_converged) CALL SCALAR_TRANSPORT( u ) CALL SCALAR_TRANSPORT( v ) CALL SCALAR_TRANSPORT( w ) CALL MASS_CONSERVATION END DO In many comressible-flow codes the main fluid variables are assembled and solved as a vector (ρ, ρu, ρv, ρw, ρe). This is called a couled aroach. 5.2 Pressure-Velocity Couling Question 1. How are velocity and ressure linked? Question 2. How does a ressure equation arise? Question 3. Should velocity and ressure be co-located (stored at the same ositions)? Link Between Pressure and Velocity The discretised momentum equation for one cell is of the form area A w u P e (2) Hence, where (3) Answer 1 (a) The force terms in the momentum equation rovide a link between velocity and ressure. (b) Velocity deends on the ressure gradient or, when discretised, on the difference between ressure values ½ cell either side. Symbolically: where indicates a centred difference ( right minus left ). Substituting for velocity in the continuity equation, CFD 5 2 David Asley W w N n P s S e E

3 This has the same algebraic form as the scalar-transort equation. Answer 2 The momentum equation rovides a link between velocity and ressure which, when substituted into the continuity equation, gives an equation for ressure. Hence, a ressure equation arises from the requirement that solutions of the momentum equation be mass-consistent Co-located Storage of Variables Suose ressure and velocity are co-located (stored at the same ositions) and that advective velocities (cell-face velocities in the mass fluxes) are calculated by linear interolation. In the momentum equation the net ressure force involves i-1 i i+1 w e Hence, the discretised momentum equation has the form: In the continuity equation the net outward mass flux deends on i-2 i-1 i i+1 i+2 w e So, both mass and momentum equations only link ressures at alternate nodes. Thus, the combination of: co-located u, ; linear interolation for advective velocities; leads to decouling of odd nodal values 1, 3, 5, from even nodal values 2, 4, 6,. This odd-even decouling or checkerboard effect leads to indeterminate oscillations in ressure. There are two common remedies: x CFD 5 3 David Asley

4 (1) use a staggered grid (velocity and ressure stored at different locations); or (2) use a co-located grid but Rhie-Chow interolation for the advective velocities. Both rovide a link between adjacent ressure nodes, reventing odd-even decouling Staggered Grid (Harlow and Welch, 1965) In the staggered-grid arrangement, velocity comonents are stored half-way between the ressure nodes that drive them. u v u ressure/scalar control volume This leads to different sets of control volumes: one set for ressure (and other scalars) and others for the different velocity comonents. v u control volume v control volume One indexing convention (esecially in comuter code) gives each velocity node the same index (P or ijk) as the ressure node to which it oints. Another (used in theoretical work) refers to, e.g., the u (i 1/2) velocity. On a cartesian mesh In the momentum equation, ressure is stored at recisely the oints required to comute the ressure force. u i-1 i i In the continuity equation velocity is stored at recisely the oints required to comute mass fluxes. For net mass flux: u i u i+1 i-1 i i+1 No interolation is required for cell-face values and there is a strong linkage between successive ressure nodes, avoiding odd-even decouling. Advantages No interolation required; variables are stored exactly where they are needed. No odd-even ressure decouling. Disadvantages Added geometrical comlexity from multile sets of nodes and control volumes. If the mesh is not cartesian then the velocity nodes may cease to lie between the ressure nodes that drive them (see right). Very difficult to imlement on unstructured meshes. CFD 5 4 David Asley

5 5.2.4 Rhie-Chow Velocity Interolation (Rhie and Chow, 1983) The alternative aroach uses co-located ressure and velocity but emloys a different, ressure-deendent, interolation for advective velocities (cell-face velocities in the mass fluxes). W P E w e The momentum equation connects velocity and ressure: Symbolically, is called the seudovelocity. It reresents everything on the RHS excet ressure. In the Rhie-Chow algorithm this symbolic relation is alied at both nodes and faces. (4) (5) (i) Invert to work out at nodes: with centred difference Δ from interolated face values (ii) Then linearly interolate and d to cell faces: with centred difference Δ taken from adjacent nodes This amounts to adding and subtracting centred ressure differences worked out at different laces; e.g. on the east face, with an overbar denoting linear interolation to that face, (6) Using this interolative technique, mass conservation gives: (7) Notes. The central ressure value does not cancel, so there is no odd-even decouling. This is a ressure equation. Moreover, it has the same algebraic form as a scalartransort equation:, where In ractice it is common to solve iteratively for ressure corrections rather than ressure itself (Section 5.3). Equation (7) then becomes where * denotes current value of. This can be rearranged as: (8) CFD 5 5 David Asley

6 Examle For the uniform cartesian mesh shown below the momentum equation gives a velocity/ressure relationshi for each cell, where Δ denotes a centred difference. f u = = A For the u and values given, calculate the advective velocity on the cell face marked f: (a) (b) (c) by linear interolation; by Rhie-Chow interolation, if A = 0.6 (local ressure maximum to the left of face); by Rhie-Chow interolation, if A = 0.9 (constant ressure gradient). Answer (a) By linear interolation, (b) The working for Rhie-Chow may be set out as follows. u by interolation of by interolation of 2.8 Answer: u f = 2.8 This is higher than that obtained from linear interolation, showing the velocity field trying to alleviate the local ressure maximum just to the left of the face. (c) If A = 0.9 then Rhie-Chow interolation roduces u f = 2.5, as for linear interolation. CFD 5 6 David Asley

7 Analysis of Rhie-Chow Interolation Parts (a) and (c) of the examle above show that Rhie-Chow interolation gives the same result as linear interolation if there is a constant ressure gradient, whereas if (as in art(b)) there is a local ressure eak to one side of the face then the advective velocity increases to comensate. For general values of u i and i (and a uniform mesh and a constant coefficient d), Rhie-Chow interolation gives (exercise): Thus, Rhie-Chow interolation adds a ressure-smoothing term to the art obtained simly by interolation. Pressure-velocity couling is the dominant feature of the Navier-Stokes equations. Staggered grids are an effective way of handling it on Cartesian meshes. However, for non-cartesian meshes, co-located storage is the norm; the Rhie-Chow algorithm is emloyed in most general-urose CFD codes. A generalised form of the Rhie-Chow algorithm (based on local ressure gradients) is used on unstructured meshes. CFD 5 7 David Asley

8 5.3 Pressure-Correction Methods Consider how changing ressure could be used to enforce mass conservation. Net mass flux in. Increase cell ressure to drive mass out. Net mass flux out. Decrease cell ressure to suck mass in. What are ressure-correction methods? Iterative numerical schemes for ressure-linked equations. Seek to roduce velocity and ressure satisfying both mass and momentum equations. Consist of alternating udates of velocity and ressure: solve the momentum equation for velocity with the current ressure; use velocity-ressure link to rehrase continuity as a ressure-correction equation; solve for ressure corrections to nudge velocity towards mass conservation. There are two common schemes: SIMPLE and PISO. Velocity and Pressure Corrections The momentum equation connects velocity and ressure: where ½ and +½ indicate the relative locations as multiles of the grid sacing. One must correct velocity to satisfy continuity: but simultaneously correct ressure so as to retain a solution of the momentum equation: The velocity-correction formula is, therefore, (9) CFD 5 8 David Asley

9 Classroom Examle 1 (Patankar, 1980) A 1 B 2 C 3 In the steady, one-dimensional, constant-density situation shown, the ressure is stored at locations 1, 2 and 3, whilst velocity u is stored at the staggered locations A, B and C. The velocity-correction formula is where ressure nodes i 1 and i lie on either side of the location for u. The value of d is 2 everywhere. The inflow boundary condition is u A = 10. If, at a given stage in the iteration rocess, the momentum equations give and, calculate the values of,, and the resulting corrected velocities. Classroom Examle 2 (Exam 2006 art) In the 2-d staggered-grid arrangement shown below, u and v (the x and y comonents of velocity), are stored at nodes indicated by arrows, whilst ressure is stored at the intermediate nodes A D. The grid sacing is uniform and the same in both directions. The velocity is fixed on the boundaries as shown. The velocity comonents at the interior nodes (u B, u D, v C and v D ) are to be found. At an intermediate stage of calculation the internal velocity values are found to be u B = 11, u D = 14, v C = 8, v D = 5 whilst correction formulae derived from the momentum equation are with geograhical (w,e,s,n) notation indicating the relative location of ressure nodes. (a) Show that alying mass conservation to control volumes centred on ressure nodes leads to simultaneous equations for the ressure corrections. Solve for the ressure corrections and use them to generate a mass-consistent flow field. C 5 10 u D D vc vd A u B B 5 5 y x (b) Exlain why, in ractice, it is necessary to solve for the ressure correction and not just the velocity corrections in order to satisfy mass conservation. CFD 5 9 David Asley

10 5.3.1 SIMPLE: Semi-Imlicit Method for Pressure-Linked Equations (Patankar and Salding, 1972) Stage 1. Solve the momentum equation with current ressure. START The resulting velocity generally won t be mass-consistent. Stage 2. Formulate the ressure-correction equation. (i) Relate changes in u to changes in : solve momentum equations solve ressurecorrection equation (ii) Make the SIMPLE aroximation: neglect. correct velocity and ressure (iii) (Legitimate, since corrections vanish in the final solution.) Aly mass conservation to control volumes centred on the ressure nodes. The net mass flux results from current (u * ) lus correction (u) velocity fields: no converged? yes END Hence, (minus the current net mass flux) or, writing in terms of the ressure correction (staggered or non-staggered mesh): This results in a ressure-correction equation of the form: (10) Stage 3. Solve the ressure-correction equation The discretised ressure-correction equation (10) has the same form as the discretised scalar-transort equation, and hence the same solver may be used. Stage 4. Correct ressure and velocity: (11) Iterate. Reeat stages 1 to 4 until mass and momentum equations are simultaneously satisfied. CFD 5 10 David Asley

11 Notes. The source term for the ressure-correction equation is minus the current net mass flux ( ), as exected. If at any stage there is net mass flow into a control volume then the ressure there must rise in order to ush mass back out. In ractice, substantial under-relaxation of the ressure udate is needed to revent divergence. In the correction ste the ressure (but not the velocity) udate is relaxed: Tyical values of α are Velocity is under-relaxed in the momentum transort equations, but the under-relaxation is generally less severe: α u As they involve the velocities being comuted, matrix elements change at each iteration. There is little to be gained by solving matrix equations exactly at each stage, but only doing enough iterations of the matrix solver to reduce the residuals by a sufficient amount. Alternative strategies at each SIMPLE iteration are: (1) m iterations of each u, v, w equation, followed by n iterations of the equation (tyically m = 1, n = 4); or (2) do enough iterations of each equation to reduce the residual error to a small fraction of the original (say 10%) Variants of SIMPLE The SIMPLE scheme can be inefficient and requires considerable ressure under-relaxation. This is because the corrected fields are good for udating velocity (since a mass-consistent flow field is roduced) but not ressure (because of the inaccuracy of the aroximation connecting velocity and ressure corrections). To remedy this, a number of variants of SIMPLE have been roduced. SIMPLER (Patankar, 1980) This variant acknowledges that the correction equation is good for udating velocity but not ressure, and recedes the momentum and ressure-correction equations with the equation for the ressure itself (equation (7)). SIMPLEC (Van Doormaal and Raithby, 1984) This scheme seeks a more accurate relationshi between velocity and ressure changes. From the momentum equations, the velocity and ressure equations are related by (12) The SIMPLE aroximation is to neglect the term (*). However, this is actually of comarable size to the LHS. In the SIMPLEC scheme, (12) is rewritten by subtracting from both sides: Assuming that, it is more accurate to neglect the term (**), thus CFD 5 11 David Asley

12 roducing an alternative formula connecting velocity and ressure changes: (13) Comare: (14) Although concetually aealing, there is a difficulty. The sum-of-the-neighbouringcoefficients constraint on a P means that, in steady-state calculations,, and hence the denominator of (13) vanishes. This roblem doesn t arise in time-varying calculations, where a time-deendent art is added to a P, removing the singularity. SIMPLEX This scheme assumes that velocity and ressure corrections are linked by some general relationshi This includes both SIMPLE and SIMPLEC as secial cases (see equation (14)). However, the SIMPLEX scheme attemts to imrove on this by actually solving equations for the P. These equations are derived from (12) but we shall not go into the details here. This author s exerience is that SIMPLER and SIMPLEX offer substantial erformance imrovements over SIMPLE on staggered grids, but that the advanced schemes are difficult (imossible?) to formulate on co-located grids and offer little advantage there. (15) PISO PISO Pressure Imlicit with Slitting of Oerators (Issa, 1986). This was originally roosed as a time-deendent, non-iterative ressure-correction method. Each timeste (t old t new ) consists of a sequence of three stages: (1) Solution of the time-deendent momentum equation with the t old ressure. (2) A ressure-correction equation and ressure/velocity udate à la SIMPLE to roduce a mass-consistent flow field. (3) A second corrector ste to roduce a second mass-consistent flow field but with timeadvanced ressure. Aart from time-deendence, stes (1) and (2) are essentially the same as SIMPLE. However, ste (3) is designed to eliminate the need for outer iteration at each time ste as would be the case with SIMPLE. However, the ressure-correction equations (2) and (3) must be solved to a much finer tolerance on each ass, so requiring many more inner iterations of the matrix equations. Evidence suggests that PISO can be more efficient in some time-deendent calculations, but SIMPLE and its variants are better in direct iteration to steady state. CFD 5 12 David Asley

13 Summary Each comonent of momentum satisfies its own scalar-transort equation: concentration, velocity comonent (u, v or w) diffusivity, Γ viscosity, μ source, S forces However, the momentum equations are: non-linear; couled; required also to be mass-consistent. and, as a consequence, have to be solved: iteratively; together; in conjunction with the mass equation. For incomressible flow, the requirement that solutions of the momentum equation be mass-consistent generates a ressure equation. Pressure-gradient source terms lead to odd-even decouling when all variables are colocated (stored at the same nodes). This may be remedied by using either: a staggered velocity grid; a non-staggered grid, but Rhie-Chow interolation for advective velocities. Pressure-correction methods make small corrections to ressure in order to nudge the velocity field towards mass conservation whilst still reserving a solution of the momentum equation. Widely-used ressure-correction algorithms are SIMPLE (and its variants) and PISO. The first is an iterative scheme; the second is a non-iterative, time-deendent scheme. References Harlow, F.H. and Welch, J.E., 1965, Numerical calculation of time-deendent viscous incomressible flow of fluid with a free surface, Physics of Fluids, 8, Issa, R.I., 1986, Solution of the imlicitly discretised fluid flow equations by oerator slitting, J. Comut. Phys., 62, Patankar, S.V., 1980, Numerical Heat Transfer and Fluid Flow, McGraw-Hill. Patankar, S.V., 1988, Ellitic systems: finite difference method I, Chater 6 in Handbook of Numerical Heat Transfer, Minkowycz, W.J., Sarrow, E.M., Schneider, G.E. and Pletcher, R.H. (eds.), Wiley. Raithby, G.D. and Schneider, G.E., 1988, Ellitic systems: finite difference method II, Chater 7 in Handbook of Numerical Heat Transfer, Minkowycz, W.J., Sarrow, E.M., Schneider, G.E. and Pletcher, R.H. (eds.), Wiley. Rhie, C.M. and Chow, W.L., 1983, A numerical study of the turbulent flow ast an isolated airfoil with trailing edge searation, AIAA Journal, 21, Van Doormaal, J.P. and Raithby, G.D., 1984, Enhancements of the SIMPLE method for redicting incomressible fluid flows, Numerical Heat Transfer, 7, CFD 5 13 David Asley

14 Examles Q1. For the rectangular control volume with surface ressures shown, what is: (a) the net force in the x direction? (b) the net force in the x direction, er unit volume? (c) the average ressure gradient in the x direction? w A x e (0,2) =9 (2,4) =5 =7 (4,0) Q2. The figure left shows a triangular cell in a 2-d unstructured mesh, together with the coordinates of its vertices and the average ressures on the cell faces. Calculate the x and y comonents of the net ressure force on the cell (er unit deth). Q3. (Exam 2014 art) The figure right shows a quadrilateral cell in a 2-d mesh, together with the coordinates of its vertices and the average ressures on the cell faces. Calculate the x and y comonents of the net ressure force on the cell (er unit deth). (1,4) =4 =10 (5,3) =2 (-1,1) =1 (4,0) Q4. (a) (b) (c) Exlain (briefly) how an equation for ressure is derived in finite-volume calculations of incomressible fluid flow. Exlain how the roblem of odd-even decouling in the discretised ressure field arises with co-located storage and linear interolation for advective velocities, and detail the Rhie-Chow interolation which can be used to overcome this. The figure below shows art of a Cartesian mesh with the velocity u and ressure at the centre of 4 control volumes. If the momentum equation leads to a ressurevelocity linkage of the form, (where Δ reresents a centred difference) use the Rhie-Chow rocedure to find the advective velocity on the cell face marked f. f u = = (d) Describe the SIMPLE ressure-correction method for the solution of couled mass and momentum equations. CFD 5 14 David Asley

15 Q5. (Exam 2016) (a) By considering an infinitesimal cuboid cell with edges aligned with the coordinate axes, show that the ressure force er unit volume in the x direction is. (b) For what urose is the Rhie-Chow algorithm used in ressure-based finite-volume simulations of incomressible fluid flow? Part of a uniform structured mesh is shown in the figure below, along with the cell-centred values of the x-velocity comonent u and the ressure in consistent units. From the discretised momentum equation, velocities and ressures are found to be linked by where Δ here denotes a centred difference. i-2 w e i-1 i i+1 i+2 = u = (c) (d) Find the velocity on the w and e faces of cell i (ositions i ½ and i + ½) using: (i) linear interolation; (ii) Rhie-Chow interolation. Comment briefly on your results and the effect of the ressure field on advective velocities. Assuming flow only in the x direction, and using the face velocities found in art (c)(ii) as the current velocity values, set u but do not solve the ressurecorrection equation for cell i. Q6. The figure defines the relative osition of velocity ( ) and ressure () nodes in a 1-d, staggered-grid arrangement. u 1 u 2 u3 u Velocity u 1 = 4 is fixed as a boundary condition. After solving the momentum equation the velocities at the other nodes are found to be u 2 = 3, u 3 = 5, u 4 = 6. The relationshi between velocity and ressure is found (from the discretised momentum equation) to be of the form at each node, where Δ denotes a centred difference in sace. Aly mass conservation to cells centred on scalar nodes, calculate the ressure corrections necessary to enforce continuity, and confirm that a mass-consistent velocity field is obtained. CFD 5 15 David Asley

16 Q7. (Exam 2013) (a) Exlain, without mathematical detail, how ressure-correction methods may be used to solve the couled equations of incomressible fluid flow. (b) State the two main differences between the PISO and SIMPLE ressure-correction methods. In the 2-d staggered-grid arrangement shown right, u and v (the x and y comonents of velocity), are stored at nodes indicated by arrows, whilst ressure is stored at intermediate nodes A, B, C, D. Grid sacing is uniform and the same in both directions. Velocity is fixed at inflow. Uer and lower boundaries are imermeable. 6 6 A v A B u C ud C v C D u E uf At an intermediate stage of calculation the outflow velocities are u E = 12, u F = 4, inflow outflow whilst the internal velocity comonents are u C = 2, u D = 1, v A = 1, v C = 2. Velocity-ressure correction formulae are with geograhical (w,e,s,n) notation indicating the relative location of ressure nodes. (c) (d) Aly a uniform scale factor to the outflow velocities to enforce global mass conservation, stating the scale factor and outflow velocities after its alication. Show that alying mass conservation to control volumes centred on ressure nodes leads to simultaneous equations for the ressure corrections. Solve for the ressure corrections and use them to generate a mass-consistent flow field. CFD 5 16 David Asley

17 Q8. (Exam 2011) (a) In a finite-volume CFD calculation, flow variables are usually stored at the centre of comutational cells. Exlain briefly, and without mathematical detail, (i) how ressure-correction methods can be used to generate a mass-consistent velocity field; (ii) why simle linear interolation may cause difficulties with the couling of ressure and velocity if these variables are stored at the same locations. (b) Part of one line of cells in a uniform Cartesian mesh is shown in the figure below, together with the velocity u and ressure at nodes. In each cell the momentum equation gives a relationshi between velocity and ressure of the form where Δ denotes a centred difference. f u = = L Calculate the velocity on the cell face marked f by Rhie-Chow interolation if: (i) L = 0.5; (ii) L = 0.3. (c) The figure below shows a triangular cell in a 2-d mesh for an inviscid CFD calculation. The vertex coordinates and the average ressures on the cell edges are shown in the figure. The density ρ = 1.0 everywhere. Find the x and y comonents of: (i) the net ressure force on the cell; (ii) the fluid acceleration. (3,5) y = 2 = 5 x (0,0) = 3 (4,0) CFD 5 17 David Asley