2.8. MMS04: No-slip wall BC, single-phase, 3D, curl-based functions

2.8.1. Description

The no-slip wall boundary condition in MFIX is verified using the techniques described in [5]; the manufactured solution is selected such that it satisfies both the divergence-free constraint and the no-slip wall boundary condition. Specifically, the no-slip wall boundary condition requires that the velocity at the (stationary) no-slip wall is zero. The manufactured solution is generated using the curl-based derivation to ensure divergence-free velocity fields [5] along with the technique given in [2] to ensure that the velocity component functions approach the value of zero at each boundary tested. The manufactured solution for the velocity field used for the verification of no-slip wall is given as [6]:

(2.46)\[\overrightarrow{V} = S^{2}\left( \overrightarrow{\nabla} \times \overrightarrow{H} \right) + 2S\left( \nabla S \times \overrightarrow{H} \right)\]

where, \(\overrightarrow{V}\) is the velocity field vector, \(S\) is the mathematical equation of the boundary being tested, and \(\overrightarrow{H}\) is a general vector field consisting of sinusoidal expressions. The manufactured solution for pressure is selected as in Eq.6.1 since there are no constraints on pressure with this boundary condition.

2.8.2. Setup

This case is setup for single-phase flows on a domain with unit dimensions; the boundary tested is the West boundary (i.e., \(x = 0\)).

Table 2.16 MMS-04 Setup, Initial and Boundary Conditions.

Computational/Physical model
3D, Steady-state, incompressible
Single-phase (no solids)
No gravity
Thermal energy equations are not solved
Turbulence equations are not solved (Laminar)
Non-uniform mesh
Central scheme
Geometry
Coordinate system	Cartesian
Domain length, \(L\) (x)	1.0	(m)
Domain height, \(H\) (y)	1.0	(m)
Domain width, \(W\) (z)	1.0	(m)
Material †
Fluid density, \(\rho_{g}\)	1.0	(kg·m-3)
Fluid viscosity, \(\mu_{g}\)	1.0	(Pa·s)
Initial Conditions
Pressure (gauge), \(P_{g}\)	MMS	(Pa)
Fluid x-velocity, \(u_{g}\)	5.0	(m·s-1)
Fluid y-velocity, \(v_{g}\)	5.0	(m·s-1)
Fluid z-velocity, \(w_{g}\)	5.0	(m·s-1)
Boundary Conditions ‡
West boundary	No-slip wall
All other boundaries	Mass inflow

† Material properties selected to ensure comparable contribution from convection and diffusion terms.

^‡ The manufactured solution is imposed on all boundaries (i.e., Dirichlet specification).

2.8.3. Results

Numerical solutions were obtained using the Central discretization scheme for 8x8, 16x16, 32x32, 64x64, and 128x128 grid meshes. Iterative convergence was not achieved for this case when pressure was solved. Hence, the pressure variable (\(P_{g}\)) was fixed by specifying pressure using the manufactured solution in the initial conditions routine and discarding the pressure solution in the main solver routine. The observed order of accuracy matches the formal order as shown in Fig. 2.16 for the velocity variables.

Fig. 2.16 Observed orders of accuracy for no-slip wall verification (3D, single-phase flows) using \(\mathbf{L}_{\mathbf{2}}\) and \(\mathbf{L}_{\mathbf{\infty}}\) norms of the discretization error.