Understanding DPMFoam/MPPICFoam

Transcription

1 Understanding DPMFoam/MPPICFoam Jeroen Hofman March 18, 2015 In this document I intend to clarify the flow solver and at a later stage, the article-fluid and article-article interaction forces as imlemented in DPMFoam in OenFoam 2.3.x. 1 The Flow Solver In this section we first set u the incomressible Navier-Stokes equations for the gas flow solver, then we describe the derivation of the ressure equation in semi-discretized form, which is used in the PIMPLE algorithm emloyed by OenFOAM in the last section. 1.1 Incomressible Navier-Stokes We start with the incomressible Navier-Stokes equation for laminar flow with gas volume fraction ɛ, constant viscosity µ, gas density ρ, gravity g, stress tensor τ and momentum transfer term F: ρ t + ρ ɛuu) ρ ɛτ = + ρɛg F 1) where the stress tensor with unit tensor δ and kinematic viscosity ν = µ ρ is given by: τ = ν ) u + u) T + 2 ν u) δ 2) 3 The interhase momentum transfer term F contains both momentum transfer due to drag and buoyancy [2], details will be given in section 1.2. Therefore it is not fully a source term since the drag is roortional to u. When dividing by the density and defining the kinematic ressure as P = ρ we obtain: t + ɛuu) ɛτ = P + ɛg F ρ 3) 1

2 We also define the continuity equation as: ɛ + ɛu) = 0 4) t 1.2 Momentum transfer In this section we look at different ways of defining momentum transfer to either the article or the fluid hase. By switching from the discrete article descrition to a continuum descrition we try to link the different ways of defining the momentum transfer term since regardless of definition of F equation 3 the resulting PDEs should be equal. We will first look at the way the momentum transfer term is usually defined as a combination of buoyancy and drag for articles in a comutational cell: F = f = 1 V f drag V i ) = F drag 1 V V i 5) where V is the volume of the comutational cell, V the volume of the article and i the locally averaged ressure gradient. In a continuum descrition we can write 1 V V i = 1 ɛ) and rewrite equation 3: t + ɛuu) ɛτ = ɛ + ɛg F drag ρ 6) Another way of incororating buoyancy in stead of via the ressure gradient is by exlicitly adding a local acceleration term lus an adjustment to the drag [6], section 2.5.1): F = f = 1 V fdrag where [ ] Dt gravity field is uniform and we use 1 V ɛ ρv g )) = F drag + 1 Dt ɛ V ρv g is the locally averaged fluid velocity at the location of article. term and we can write equation 3 as: [ ] ) Dt Using that the ρv g = 1 ɛ)ρg we take this term out of the interaction 7) t + ɛuu) ɛτ = P + g F drag ɛρ 1 V [ ] V Dt 8) Furthermore realizing that, by using continuity in equation 4 and using roduct rules, we have: t + ɛuu) = ɛ Dt = ɛ u + ɛu u 9) t 2

3 and using that in the continuum descrition we have 1 V second form of the momentum equation: V Dt ɛτ = + g F drag ɛρ [ ] Dt = 1 ɛ) Dt we obtain a 10) Exressions 6 and 10 should be the same when the latter is multilied by ɛ, however there are clearly not because of the treatment of the stress term. I have sent quite some time digging into this roblem and it turns out this is a very subtle roblem arising from the recise definitions of both fluid and solid equations in a continuum descrition. See for instance [7], where the authors show that many well-cited) aers even mess this u. Let us first consider how to fix this roblem: 1. Omitting the viscuous stress is the aroach that is taken in the MP-PIC aers [8], this will make equations 6 and 10 equal. Arguably this is valid as the driving factors are often drag and article ressure, and esecially for gases the viscosity is very small. 2. The aroach alied in both [7] and [6] is changing the ansatz of the derivation, namely the momentum equation; equation 3. Similar to accounting for ressure in both the gas hase momentum equation and the article momentum transfer term, we can instead also slit the viscuous stress in this way. We then obtain: t + ɛuu) τ = P + ɛg F 11) ρ F = F drag 1 V i + τ i ) 12) V Combining the above equations and using the continuum descrition we end u with: ɛ Dt ɛ τ = ɛ + ɛg F drag ρ 13) Similarly if we use equation 11 we can follow the same stes when incororating buoyancy via the local acceleration as we did before and then we obtain the following relacement of equation 10: Dt τ = + g F drag ɛρ 14) Now we see that the aroaches do line u. Some comments can be made here: first of all in this aroach we end u with the hase fraction ɛ outside the divergence term of the viscuous stress. Arguably this is valid when ɛ 0 since ɛτ = ɛ τ + τ ɛ, however this is clearly not the case in for instance fluidized beds. Furthermore a simle derivation using control volumes and flux evaluation will not give you an answer to this question. Instead, one can theoretically derive the momentum equations in a continuum framework) in which there are two aroaches [9], chater 3): the Jackson aroach, in which the viscuous stress 3

4 has ɛ outside the divergence term, but also includes 1 ɛ) τ in the article momentum equations; and secondly the Ishii aroach, in which ɛ is inside the divergence term of the vicuous stress but not included in the article equations. These aroaches also known as model A and model B in literature) cause a high amount of confusion and often are wrongly imlemented in for instance TFM, however the difference can be subtle as again viscuous stresses are deemed not so imortant. If we omit viscuous stress the models are the same. The confusion is further enhanced when the article hase is discrete, since we introduce an extra layer of comlexity. Returning to the key-oint of this reort, namely understanding OenFOAM, it seems to use a mixed aroach. The momentum equation is as equation 3 and it uses the locally averaged fluid velocity as substitute for the ressure gradient, so eventually equation 8 is used, where the drag F drag is defined as: 1 1 ɛ)v V β u u ) 15) where β is defined according to the Ergun-Wen-Yu drag model [9], u is the article velocity and u is the fluid velocity interolated to the article osition. 1.3 Discretization There are many choices regarding discretization of the terms in equation 3, which we will not discuss in deth here. We only highlight that when looking at the convection term ɛuu) we see that the discretized system would be non-linear. A ossible solution would be to solve this system using non-linear solvers but this is in general quite exensive and imractical. A second solution would be to linearize the convection term. Using the discretized version of Gauss theorem [1] we can write for a control volume V P with faces f centered around a oint P, where all variables are defined at cell centres: V P ɛuu)dv = f = f = f S ɛuu) f S ɛu) f u f F u f 16) where we have defined F = S ɛu) f as the hase) mass face flux and values are assumed to be interolated from cell centers 1. F has to satisfy continuity, which is equation 4. 1 We make the assumtion that ab) f = a f b f, which is not necessarily true for interolated values, but often assumed to make interolations easier, see for instance the discussion in [3] 4

5 By linearizing the convection term it follows that we define the fluxes F as a time-lagged quantity, i.e. we assume we have an existing velocity flux field F satisfying equation 4. We can now write the discretized version of the convection term as a linear combination of the velocity: V P ɛuu)dv = f F u f u P + N a N u N 17) where the coefficients a N and contain the known fluxes F and the velocity at the face is assumed to be interolated from the control volume and neighboring control volumes. This imlicitly also assumes the hase fraction ɛ to be known when discretizing u. 1.4 Derivation of the ressure equation Since equation 3 can be discretized in a linear fashion following the treatment of the convection term above, we can semi-discretize the momentum equation. F can be slit as a combination of a contribution to the velocity and a source term, i.e. F = f P u P + f. Details of the slit were discussed in section 1.2. For now it is sufficient that the slit can be made. The semi-discretized equation has the following form, combining the diagonal coefficients in one coefficient : a P f ) P u P = ρ N a N u N + a 0 u 0 P + g + f ρ P u P = Hu) P u P = Hu) P 18) where the subscrit 0 indicates the revious value. Hu) contains all off-diagonal contributions of the linear system, including source terms. The semi-discretized version of the continuity equation equation 4) can be written as: ɛ t + f S ɛu) f = 0 19) where we assume some exlicit discretization of the temoral term, not shown here. We first exress the velocity of the cell face as: ) Hu) u f = f ) P 20) where P, Hu) and are interolated to the cell faces. Interolation of P is erformed by simle differencing of both control volumes adjacent to the face, i.e. P ) f P P P E x, where control volumes with centres P and E are adjacent to face f and x is the distance between P and E. By 5

6 directly calculating the gradient of ressure on the cell faces by the ressure in adjacent cells and not interolating the gradient itself, checkerboard roblems for the ressure are avoided. By combining equation 19 and 20 we obtain the ressure equation: f S ɛ f P ) f = ɛ t + f ) Hu) S ɛ f 21) The flux F is defined as: [ Hu) ) F = S ɛu) f = S ɛ f f ) ] P 22) This flux is guaranteed to be conservative by construction. 1.5 Rhie-Chow interolation It is well-known that solving the incomressible Navier-Stokes equations on a collocated grid gives rise to the so-called checkerboard roblem when using central differencing, where the solutions for velocity and ressure field are no longer unique u to a constant in the case of ressure), see [4], chater 7.5. A well-known cure to this roblem is the so called Rhie-Chow interolation rocedure, where the face velocity is adjusted as: ) 1 [ ] u f = u f A Pf P f 23) The overline denotes interolated values. In other words, the interolated velocity is corrected by the difference in the gradient of the ressure at the cell face and the interolated gradient. It can be shown that this correction is roortional to the third derivative of the ressure, corresonding to a fourth order correction in the continuity equation. We will now show something similar haens when deriving the ressure equation as we did in the revious section. Consider again equation 18 where we exlicitly state that the gradient of ressure is evaluated at oint P : u P = Hu) P ) P 24) By defining the cell-faced velocity as in equation 20 and combining with the above we have: u f = u P + P ) ) ) P P 25) Writing out the interolation oeration with a f = a E ) we get an exression that is very similar to equation 23: 6

7 u f = 1 2 u P + u E ) + 1 [ P ) P + P ) E ] 1 P P P E 2a f a f x = 1 2 u P + u E ) + 1 [ PEE P P + P ] E P W 1 P P P E 2a f 2 x 2 x a f x = 1 2 u P + u E ) + 1 4a f x [P EE 3P E + 3P P P W ] 26) the last term on the right hand side of the above equation is indeed the discretization of the third derivative of the ressure, the same as the Rhie Chow interolation. 1.6 PIMPLE in OenFOAM We are now ready to describe the combined PISO + SIMPLE algorithm that is used in OenFOAM. The following stes can be identified: 1. Set u the discretized equation for u on the grid without any source terms, i.e. we set u a coefficient matrix according to the first line of equation 18, excluding the source terms. 2. Relax this equation. 3. Solve equation 18 with an initial ressure field estimate P guess to obtain the momentum redictor u at the cell centres. P guess is usually chosen equal to the ressure field in the revious time-ste. 4. Using the redictor u we can assemble the oerator Hu ) and interolate both and Hu ) to the cell faces. 5. Solve the ressure equation equation 21) on the cell faces. 6. Udate the flux field at the cell faces using the obtained ressure equation 22). 7. Relax the ressure. 8. Udate the velocity at the cell centres using the obtained ressure equation 18). 9. Udate the boundary conditions for consistency. Stes 1-9 can be reeated any number of times. This, together with the relaxation, is the essence of the SIMPLE algorithm [5]. The momentum correction stes 4-9) can also be reeated any number of times, this is the essence of the PISO algorithm [5]. OenFOAM defines the PIMPLE SIMPLE + PISO) algorithm alying both loos, including relaxation. Combining the methods in general has roven to give faster convergence. 7

8 2 Particle contributions in MPPICFoam 2.1 Particle-Particle forces 3 MPPICFoam/DPMFoam code 3.1 Parcel Evolution 3.2 Flow Solving References [1] htt://owerlab.fsb.hr/ed/kturbo/oenfoam/docs/hrvojejasakphd.df [2] Andrews, M.J., O Rourke, P.J., 1996, The multihase article-in-cell MP-PIC) method for dense articulate flows, Int. J. Multihase Flow 22, [3] htt:// [4] Ferziger, J. H. and Peric, M., Comutational Methods for Fluid Dynamics, 2nd ed., Sringer- Verlag 2001) [5] Versteeg H.K., Malalasekera W., An introduction to comutational fluid dynamics, the finite volume method 2nd ed.), [6] htt://gradworks.umi.com/33/03/ html [7] Z. Y. ZHOU, S. B. KUANG, K. W. CHU and A. B. YU 2010). Discrete article simulation of article-fluid flow: model formulations and their alicability. Journal of Fluid Mechanics, 661, doi: /s x [8] O Rourke, P.J., Snider, D.M., 2010, An imroved collision daming time for MP-PIC calculations of dense article flows with alications to olydiserse sedimenting beds and colliding jet articles, Chemical Engineering Science 65, [9] Wachem, B., Derivation, imlementation, and validation of comuter simulation models for gas-solid fluidized beds, PhD thesis of our grou,