2.7 Discretization procedure
2.7.2 Discretization of the momentum equation
The details of the discretization of the temporal term, convective and diffusive fluxes of the momentum equations have been discussed below in details.
Temporal term
The unsteady or temporal term of the momentum equation can be discretized as Z
∆V
∂(αρφ)
∂t dV ≈ (αρφ)n+1co −(αρφ)nco
∆t ∆Vco (2.55)
where ∆Vco is the volume of the owner cell co and the cell-centroidal values of any variable φ are assumed to be equal to the cell-averaged quantities.
Treatment of the Convection Fluxes
The convective fluxes are discretized by first converting the volume integral to a surface integral and approximating the later with a single point Gauss quadrature.
Z
∆V
∇ ·(αρuφ)dV = Z
∆S
(αρuφ)·dS≈X
f
αfρfφf(u·S)f =X
f
αfρfFfφf (2.56) Here the values of ρf and αf are obtained from the cell-centered value using inverse volume-weighted interpolation. φf is the value ofφ at the center of facef. Now, in
2.7 Discretization procedure 43 order to calculate the value of Ffφf for a face f, a weighted average of upwind and central difference scheme is used following the deferred correction approach of Khosla and Rubin [98], so that the problems relating to numerical diffusion and numerical dispersion can be avoided. For a face f, the convective flux can be estimated as a linear combination of upwind and central difference scheme as given below.
Ffφf = (Ffφf)U DSn+1+γ
(Ffφf)CDSn−(Ffφf)U DSn
(2.57) where γ is the deferred constant whose value varies in between 0 and 1. Fluxes at faces due to first order upwind scheme (UDS) and central difference scheme (CDS) can be expressed in the following manner.
FfφfU DS
=φco[|Ff,0|]−φnb[| −Ff,0|] (2.58) Here [|Ff,0|] denotes the the maximum of the two arguments and φnb denotes the value of the variable φ in the corresponding neighbor cell. On the other hand, the flux at the face f using CDS can be calculated using inverse volume-weighted interpolation as given below.
FfφfCDS =Ff
∆Vnb
∆Vnb+ ∆Vco
φco+ ∆Vco
∆Vnb+ ∆Vco
φnb
(2.59) where ∆Vco is the volume of the owner cell co and ∆Vnb is the volume of the cor- responding neighbor cell nb. The first term, i.e., the upwind part at (n+ 1) time level in Eq. 2.57 is incorporated in the coefficients of the unknown velocity during pressure velocity iteration while the second part is incorporated as source term of the same equation and are evaluated using previous iteration values. The deferred correction approach enhances the diagonal dominance of the coefficient matrix to enhance the stability of the solution algorithm.
Treatment of the diffusive fluxes
The diffusive term in the momentum equation is computed using Green-Gauss re- construction which is second order accurate. Using Gauss divergence theorem, the diffusive fluxes for a general variable φ for a particular face f is calculated in the following manner.
Z
V∇ ·(µα∇φ) dV = Z
S
(µα∇φ)·dS≈X
f
(µα∇φ·S)f =X
f
Fdφf
where Sf is the face area vector with a magnitude of Af and its direction is given by unit vector nˆf. For orthogonal grid arrangement, this face area vector is parallel to the line joining the adjacent cell centers whereas for non-orthogonal grids, this face area vector is inclined to the line joining the adjacent cell centers as shown in Fig. 2.3. and thus giving rise to the non-orthogonal component of diffusive
O a
b
c a
b
c
d e
n nb f
nb nb
nf co
co co
Figure 2.3: Pictorial representation of diffusive flux calculation.
fluxes. This is explained with the help of Fig. 2.3 where the face “abc” is shared by two neighboring tetrahedral cells with centers marked as co and nb. It can be clearly understood from Fig. 2.3 that the face unit normal represented by nˆf is not parallel to the line joining the cell centers co and nb and thus for this case, along with the normal diffusion, there will be a component of diffusive flux for this non-orthogonality. Projecting the cell centers over the surface normal will result in two more points co′ and nb′. For a particular face, the diffusive fluxes for a variable φ can be calculated as,
(∇φ)f ·Sf =Af(∇φ)f ·nˆf =Af
(φnb′ −φco′)
∆n (2.60)
where ∆n is the distance between points co′ and nb′. φ′nb and φ′co are the values of φ at the points nb′ and co′ respectively which can be approximated from the values of φ at the points co and nb using Taylor series expansion, neglecting higher order terms as given below.
φnb′ =φnb+∇φnb·−−−→
nb nb′ (2.61)
φco′ =φco+∇φco·−−−→
co co′ (2.62)
2.7 Discretization procedure 45 Substituting Eqs. 2.61 and 2.62 in Eq. 2.60, finally the expression for the net diffusive flux can be found out as
(∇φ)f ·Sf =Af
(φnb−φco)
∆n +∇φnb·−−−→
nb nb′ − ∇φco·−−−→
co co′
∆n
(2.63)
The first part of Eq. 2.63 is the normal or orthogonal component of diffusive fluxes while the second part is mainly appearing due to non-orthogonality of the grids.
It is to be noted that for calculating value at the face for any variable φ, inverse volume weighted interpolation technique employed by Dalal et al. [97] is used unless otherwise stated.
Source terms and their discretization
In the momentum equations of both the phases, source terms appear and consist of pressure gradient, gravity, drag force terms in both the phases and solid pressure gradient in particle phase alone. The drag force term appearing in the momentum equations of both the phases is treated volumetrically. The gradient of solid pressure term appearing in the particle phase momentum equation is treated using standard Green Gauss approach. The numerical treatment of all the above mentioned source terms appearing in the momentum equation are discussed below.
Pressure gradient and gravity
In this work, in order to obtain a proper force balance between pressure and grav- ity, a variant of the method first proposed by Perot [99] has been adopted. In this approach, the centroidal quantities are calculated from the respective face centered values, wherein the face normal velocities are utilized to calculate the cell centroidal velocity vector. This concept has been utilized by Manik [100], but instead of cal- culating the cell centered velocity vector, this method has been successfully applied to calculate cell centroidal values of pressure gradients and other cell centered quan- tities like gravity forces for interfacial multiphase flows. This approach is termed as Modified Green Gauss (MGG) in [100] and is used for the treatment of pressure and gravity terms in this work. Hence, the pressure gradient at the cell centroid is calculated as
Z
∆V
(α∇p)dV =α∇p∆Vco=X
f
( αf
∂p
∂n
f
(xf −xco)|Sf| )
(2.64)
where the subscripts “co” and “f” denotes the values at cell center and faces re- spectively. In a similar way, the gravity term is treated as follows.
Z
∆V
(αρg)dV =αρg∆Vco=X
f
nαfρf(g·n)ˆ f(xf −xco)|Sf|o
(2.65)
In Eqs. 2.64 and 2.65,x denotes the co-ordinates of face and cell centroids and|Sf| represents magnitude of face area vector.
Gradient of solid pressure term
This term exists only in the particle phase momentum equation and the Green Gauss reconstruction is used to discretize it as follows.
Z
∆V
−∇psdV =− Z
S
psdS≈ −X
f
psfSf (2.66)
Where psf is the solid pressure at the face center of and Sf is the surface area vector of face f. For evaluating solid pressure at the face center, inverse volume interpola- tion method is used.
Drag force term
This term is present in the momentum equations of both the phases and it is treated volumetrically. This term (denoted by Sφ) is integrated over the cell volume as
Z
∆V
SφdV ≈ (Sφ)co∆Vco (2.67)