Alessandro Mauro 1,2 , Perumal Nithiarasu 1 , Nicola Massarotti 2 , and Fausto Arpino 3
4.2 The Finite Element Method
4.2.6 Characteristic-based split scheme
It is essential to understand the characteristic Galerkin procedure, discussed in Section 2.4 for the convection–diffusion equation, in order to apply the concept to solve the real convection equations. Unlike the convection–diffusion equation, the momentum equation, which is part of a set of heat convection equations, is a vector equation. A direct extension of the CG scheme to solve the momentum equation is difficult. In order to apply the CG approach to the momentum equations, we have to introduce two steps. In the first step, the pressure term from the momentum equation will be dropped and an intermediate velocity field will be calculated.
In the second step, the intermediate velocities will be corrected. This two-step procedure for the treatment of the momentum equations has two advantages. The first advantage is that without the pressure terms, each component of the momentum equation is similar to that of a convection–diffusion equation and the CG procedure can be readily applied. The second advantage is that removing the pressure term from the momentum equations enhances the pressure stability and allows the use of arbitrary interpolation functions for both velocity and pressure. In other words, the well-known Babuska–Brezzi condition is satisfied (Babuska [24], Brezzi and Fortin [25], Chung [26]). Owing to the split introduced in the equations, the method is referred to as the CBS scheme.
The CG procedure may be applied to the individual momentum components without removing the pressure term, provided the pressure term is treated as a source term. However, such a procedure will lose the advantages mentioned in the previous paragraph. For more mathematical details, please refer to Zienkiewicz et al. [20], Zienkiewicz and Codina [23], Nithiarasu [27] and Zienkiewicz et al. [28].
In order to apply the CG procedure, we can refer to the general case of governing generalized porous medium flow and heat transfer equations in non-dimensional form and indicial notation, for mixed convection, that have been presented in Section 1.3 (see equations (24)–(26)).
From the governing equations, it is obvious that the application of the CG scheme is not straightforward. However, by implementing the following procedure, it is possible to obtain a solution to the convection heat transfer porous medium equations. The solution of the free fluid flow equations is obtained by applying the same procedure.
Temporal discretization
For the sake of simplicity, the asterisks are omitted and the Darcy and Forchheimer terms in equation (25) are grouped as a “porous” term to obtain:
P=
1
ReDa+ F
√Da|u|
(72)
Temporal discretization along the characteristics of continuity, momentum and energy conservation equations results in the following set of equations:
∂uni+ϑ1
In equations (74) and (75), additional stabilization terms appear naturally from the discretization along the characteristics (Zienkiewicz et al. [20]), while the char-acteristic parameters ϑ1, ϑ2and ϑ3(0.5≤ ϑ1≤ 1 and 0 ≤ ϑi≤ 1, with i = 2, 3) are defined according to the following:
∂uni+ϑ1
Different versions of the CBS scheme can be obtained depending on the value of the above parameters. In particular an SI and an AC version of the CBS scheme can be obtained by varying the parameter ϑ2. For ϑ2between 0.5 and 1, the SI-CBS is obtained, while for ϑ2equal to 0, the AC-CBS scheme is derived. Moreover, for ϑ3between 0.5 and 1, an implicit treatment for the porous term is obtained, while for ϑ3equal to 0, an explicit one is derived.
The splitting in the CBS scheme consists of solving the above equations in a number of subsequent steps. In the first step, the pressure term is removed from equation (74) and the intermediate velocity components˜uiare obtained from:
Step 1: Intermediate velocity calculation
˜ui(1+ tεPϑ3)− uni(1+ tεP (ϑ3− 1)) = −
Removing the pressure term from the momentum equation, the pressure stability is enhanced and the use of arbitrary interpolation functions for both velocity and pressure is allowed. In other words, the well-known Babuska–Brezzi condition is satisfied (Babuska [24], Brezzi and Fortin [25], Chung [26]). The correct velocities can be determined, once the pressure field is known, using the equation:
Step 3: Velocity correction
uni+1− ˜ui = − 1 The solution of equation (78) is the third step of the algorithm.
The second step consists in the pressure calculation through the continuity equa-tion. This second step is where the SI and AC versions of the scheme differ. In particular, this second step is obtained in the SI version of the scheme by deriving equation (78) with respect to xiand imposing equation (73), obtaining the following Poisson type of equation:
Step 2 SI-CBS: Pressure calculation
0= −t1
Therefore, in this case, the incompressibility constraint is satisfied at each iter-ation, and the pressure, evaluated through equation (79), represents the actual pressure. However, the solution of equation (79) needs a matrix inversion. Further-more, the SI version of the scheme uses a global time step, which is the minimum value of the time step limit over the entire domain.
Alternatively, the AC scheme can be derived when the left hand side of equa-tion (79) is obtained from a mass conservaequa-tion equaequa-tion retaining the transient density term:
In general, it is possible to relate the density time variation to the pressure time variation, through the speed of sound, as follows:
∂ρf
∂t = 1 c2
∂pf
∂t (81)
where the real compressibility parameter, c (compressible wave speed), approaches infinity for many incompressible flow problems and the solution scheme becomes stiff and imposes severe time step restrictions. However, this parameter can be replaced locally by an appropriate artificial value, β, of finite value, employing the
AC method, that was first introduced by Chorin [29] and then further developed
In equation (82), the pressure is an artificial pressure, because the incompressibility constraint is not achieved at each time step, but only when steady-state convergence is reached. The superscripts it + 1 and it are referred to the iterative procedure and do not refer to real time levels. The second step of the AC scheme is carried out by imposing the equation (82) as:
Step 2 AC-CBS: Pressure calculation 1 The local value of β is calculated through the following procedure:
β= max (0.5, uconv, udiff, uther) (84) The local convective, diffusive and thermal velocities can be calculated through the following non-dimensional relations:
uconv=√
uiui, udiff = 2
hRe, uther= 2
hRe Pr. (85)
The diffusive time step limitation for the AC method may be written as h2/Re/2, while the convective time step limitation may be written as:
tconv= h
|u| + β (86)
The above relation includes the viscous effect via the artificial parameter.
Depending on the problem of interest, it is possible to consider other equations coupled to the above set, such as species concentration for multi-component flows.
The fourth step, for non-isothermal problems, is represented by the energy con-servation equation (75) that allows calculating the temperature at every iteration for problems with a coupling between momentum and energy conservation equations occurs.
Spatial discretization and solution methodology
The spatial discretization of the governing equations is obtained through Galerkin finite element procedure and triangular elements. Within an element, each variable is calculated through linear approximation on the basis of nodal values, according to the following equation:
φ=
3 n=1
Nnφn (87)
where Nnis the shape function at node n and φnis the value of the generic variable φ at node n. Applying the standard Galerkin procedure to the set of equations presented in the previous section, the following four steps, expressed in a matrix form, are obtained:
Step 1: Intermediate velocity calculation
M˜ui= (1 − tεP)Muin−t Step 2 SI-CBS: Pressure calculation
[Kppf]n+1 = − 1
εt[D˜ui] (89)
Step 2 AC-CBS: Pressure calculation 1
β2M(pitart+1− pitart)= −t
ε [D˜ui]− t2[Kppart]it (90) Step 3: Velocity correction
M(uni+1− ˜ui)= −εt
Step 4: Temperature calculation
MTn+1= MTn− t[CT]n− tλ
where M is the mass matrix, C is the convection matrix, Kdis the diffusion matrix, Kuis the stabilization matrix obtained from higher-order terms, fdand fuare the boundary vectors from the momentum equation, Kpis the stiffness matrix, D is the gradient matrix, ftand futare the boundary vectors from the temperature equation.
Details of all the terms presented in equations (88)–(92) are given in Lewis et al.
[19] and Zienkiewicz et al. [20].
In the present procedure, the mass matrix is lumped using a standard row-summing approach, inverted and stored in an array during the pre-processing.
Therefore, in the AC formulation, the calculation of the artificial pressure at it+ 1 iteration, through equation (90), does not need a matrix inversion and the result is a matrix inversion-free procedure. On the other hand, in the second step of the SI solution procedure, the stiffness matrix needs to be inverted at each iteration.
u = 0, v = 0, q = 1 u = 0, v = 0, q = 1 u = 1
T = 0 y
x
(b) (a)
Figure 4.31. Forced convection in a porous channel. (a) Computational domain and boundary condition; (b) structured computational grid (3,321 nodes, 6,400 elements).
Some examples
Both the schemes presented above can be used to solve many engineering problems, such as mixed, natural or forced convection both in fluid-saturated porous media and in partly porous domains, or simply in free fluid flow cases.
As mentioned before, changing the properties such as porosity and thus per-meability, it is possible to handle porous medium-free fluid interface problems as a single problem with different properties. The following limits are used for the porous medium part and for the free fluid part:
ε <1
Da= finite ⇒porous medium ε= 1
Da→ ∞ ⇒free fluid (93) A suitable set of matching conditions, to connect the porous medium and the free fluid domains, is needed (Massarotti et al. [43]).
The first example is the forced convection heat transfer inside a uniform porous channel with constant wall heat flux. The computational domain, together with the boundary conditions employed, is sketched in Figure 4.31a, while Figure 4.31b shows the computational grid employed. The flow enters the domain from the left with a constant velocity and at a non-dimensional temperature T= 0. The channel walls are heated by a constant heat flux. The numerical results have been obtained using a non-uniform grid with 6,400 elements and 3,321 nodes.
Figure 4.32 shows the non-dimensional velocity and temperature profiles, for different Darcy numbers, varying from 10−4 to 1.0 at a section, where the flow and the thermal field are hydrodynamically and thermally developed. The results have been compared with the analytical solutions presented by Nield et al. [44] and Lauriat and Vafai [45]. The dimensionless temperature is evaluated using:
Tmix = 1 umeanSx
&
Sx
uiTidS (94)
y
Figure 4.32. Forced convection in a porous channel: (a) non-dimensional velocity as function of transverse distance; (b) non-dimensional temperature as function of transverse distance.
where umeanis the mean velocity in the section considered Sx.
For a small Darcy number (10−4), the velocity profile is nearly independent of the transverse distance and slip flow occurs at the walls (Figure 4.32). An increase in Darcy number leads to a non-linear distribution of the velocity. When the Darcy number is equal to unity, the velocity profile approaches a profile similar to that of free fluid flow.
The non-dimensional temperature increases as the Darcy number decreases, as shown in Figure 4.32. A decrease in the Darcy number corresponds to a decrease in the fluid velocity in the middle of the channel and therefore the conduction heat transfer is dominating over the effect of convection.
Figure 4.33 shows the convergence histories to steady state for the AC-CBS and SI-CBS schemes obtained by using the following L2norm for velocities:
Velocity residual=
This figure shows that the SI version of the scheme converges faster as the Darcy number decreases. Instead, the AC version of the scheme has an opposite behaviour.
Table 4.1 reports the CPU time needed by both the procedures to reach steady state solution, on a machine with 4 Gb of RAM and 2.4 GHz CPU speed. These results confirm the behaviour shown in Figure 4.33. In particular, Table 4.1 shows that the AC-CBS scheme converges faster than the SI-CBS scheme when Da≥ 10−2. It was noticed that the SI-CBS scheme needs more time than the AC-CBS scheme per time iteration. In correspondence of smaller Darcy numbers (Da≤ 10−3), the AC-CBS scheme takes more time to reach the steady state. This is due to the time step
Iterations
Velocity residual
100
100 101 102 103 104 105 10−1
10−2
10−3
10−4
10−5
10−6
AC-CBS SI-CBS
(a)
Iterations
Velocity residual
100
100 101 102 103 104 105 10−1
10−2
10−3
10−4
10−5
10−6
AC-CBS SI-CBS
(b)
Figure 4.33. Convergence histories for SI and AC scheme. (a) Da= 1;
(b) Da= 10−4.
Table 4.1. CPU time (s) for forced convection in a porous channel at different Darcy numbers
Da 1 10−1 5× 10−2 10−2 10−3 10−4
AC-CBS 35 87 137 491 4,341 6,400
SI-CBS 14,730 10,250 8,055 3,357 1,107 250
calculation procedure used. Essentially, the AC scheme becomes slower in reaching the steady state when diffusion is overriding the effect of convection (Da≤ 10−3) and the local time step approaches the global time step, losing the advantage of using a higher time step in the convective zones.
The second example is the natural convection in a cavity heated uniformly from the bottom side. The computational domain, together with the boundary conditions employed, is sketched in Figure 4.34a, while Figure 4.34b shows the computational grid employed, composed of 7,200 triangular elements and 3,721 nodes.
Figure 4.35 shows the dimensionless temperature contours for a Prandtl number equal to 0.71, and for two different Rayleigh numbers (106and 7× 104) and two different Darcy numbers (10−3and 10−4). The results obtained have been compared with the numerical solution presented by Basak et al. [46].
In general, the fluid circulation is strongly dependent on the Darcy number.
In fact, when smaller Rayleigh and Darcy numbers are considered, the flow is very weak and the temperature distribution is similar to that of stationary fluid (Figure 4.35a). As the Darcy number increases, the role of convection becomes
1
Figure 4.34. Natural convection in a porous cavity heated from the bottom side:
(a) computational domain and boundary conditions; (b) structured computational grid (3,721 nodes, 7,200 elements).
0.4
Figure 4.35. Temperature contours for natural convection in a porous cavity heated from the bottom side. (a) Ra= 7 × 104and Da= 10−4; (b) Ra= 106 and Da= 10−3.
more significant and the fluid rises up strongly from the middle portion of the bottom wall, as depicted in Figure 4.35b.
Figure 4.36 shows the steady-state convergence histories for the AC-CBS and the SI-CBS schemes. This figure shows a different behaviour from that of the case of forced convection flow problem. When natural convection occurs, the AC-CBS scheme converges faster than the SI-CBS scheme for any Rayleigh or Darcy number, both in terms of number of iterations (Figure 4.36) and CPU time to reach the steady state (Table 4.2).
The SI-CBS algorithm takes more time to reach the steady state because of the coupling between momentum and energy conservation equations present in natural convection problems. The coupling is due to the presence of a generation term on
Velocity residual
Table 4.2. CPU time (s) for natural convection in a porous cavity heated from the bottom at different Rayleigh and Darcy numbers
Ra 106 7× 104
Da 10−3 10−4 10−3 10−4
AC-CBS 180 930 290 700
SI-CBS 63,270 48,230 53,400 47,560
the right hand side of the momentum conservation equation in gravity direction that must be evaluated at each time step. This term can be calculated only after the resolution of the energy equation (step 4 of the CBS algorithm), due to the fact that the nodal temperature values are unknown. This coupled system becomes stiff and causes a restriction on the global time step value used in the SI procedure and thus the solution to the simultaneous equations at the second step of the algorithm needs more time.
Moreover, the SI scheme experiences difficulties in reaching the steady state when very refined meshes are used (Massarotti et al. [47]). On the other hand, the AC-CBS scheme does not show any problem in reaching the convergence when natural convection flow occurs. The reason for this is the robust local time stepping procedure used by the AC-CBS scheme.
The last example is the developing mixed convection in a region partially filled with a fluid-saturated porous medium, confined between two vertical hot walls.
In particular, because of the geometrical symmetry, half domain has been studied.
Figure 4.37 shows the computational domain and the boundary conditions employed and the details of the computational grid near the entrance. The mesh employed has
u = v =0
(a) (b)
T = 1
v = 1, T = 0 L
y x
u = v =0 T = 1
Figure 4.37. Mixed convection in a vertical channel partially filled with a porous medium. (a) Computational domain and boundary conditions;
(b) detail of the structured computational grid near the entrance (8,591 nodes and 16,800 elements).
x
0 1
0 1 2
v/v0
y = 3
y = 32 y = 0.5
Chang [48]
AC-CBS 1.5
0.5
0.5
Figure 4.38. Mixed convection in a vertical parallel plate channel partially filled with a porous medium. Non-dimensional vertical velocity at different heights of the channel and at Da= 10−5.
8,591 nodes and 16,800 elements and is refined near the wall, near the inlet region and at the interface.
Figure 4.38 shows the dimensionless vertical velocity profile at different heights of the channel (y) for a Darcy number equal to 10−5, obtained by using the AC-CBS scheme. The present results are compared with the numerical results of Chang and
Chang [48]. The parameters used for the present investigation are Ra= 0.72 × 103, Pr= 0.72, Re = 50, λ = 2.8 in the porous region and λ = 1 in the free fluid region, ε= 0.8. When y is small, there is a large velocity difference at the interface of the composite system (Figure 4.38). When y increases, the flow discharge in the porous layer decreases, and the peak of the fluid velocity profile moves to the central axis of the vertical parallel-plate channel.
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