# 2.6. MMS02: Two-phase, 3D, curl-based functions with constant volume fraction¶

## 2.6.1. Description¶

Assuming that gas and solid volume fractions (i.e., $$\varepsilon_{g}$$ and $$\varepsilon_{s}$$) remain constant, we can see from gas and solid continuity equations that both fluid and solid velocity fields are divergence-free (for constant density of fluid and solids). A manufactured solution for the fluid-phase velocity field is defined using the curl-based approach developed in [5]. For the solid-phase velocity field, a set of simple sinusoidal functions is selected (same as those shown in Eq.2.45). The manufactured solutions for scalar quantities (pressure, gas temperature, and solid temperature) can be multivariate functions of sines and cosines as defined in Eq.6.1. The selected functions for all concerned variables are shown over a 3D domain in Fig. 2.10 through Fig. 2.12.

Fig. 2.10 Gas phase momentum equation manufactured solutions for 3D, steady-state, two-phase flow verification test case.

Fig. 2.11 Solids phase momentum equation manufactured solutions for 3D, steady-state, two-phase flow verification test case.

Fig. 2.12 Scalar field manufactured solutions for 3D, steady-state, two-phase flow verification test case.

## 2.6.2. Setup¶

Table 2.14 MMS-02 Setup, Initial and Boundary Conditions.

Computational/Physical model

Two-phase

No gravity

Drag model is turned off

Friction model is turned off

Thermal energy equations are solved

Granular energy equation is not solved

Turbulence equations are not solved (Laminar)

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)

Fluid specific heat, $$C_{\text{pg}}$$

0.05

(J·kg-1·K-1)

Fluid thermal conductivity, $$k_{g}$$

1.0

(J·kg-1·K-1·s-1)

Solids density, $$\rho_{s}$$

2.0

(kg·m-3)

Solids viscosity, $$\mu_{s}$$

2.0

(Pa·s)

Solids specific heat, $$C_{\text{ps}}$$

0.1

(J·kg-1·K-1)

Solids thermal conductivity, $$k_{s}$$

2.0

(J·kg-1·K-1·s-1)

Initial Conditions

Pressure (gauge), $$P_{g}$$

0.0

(Pa)

Fluid x-velocity, $$u_{g}$$

10.0

(m·s-1)

Fluid y-velocity, $$v_{g}$$

10.0

(m·s-1)

Fluid z-velocity, $$w_{g}$$

10.0

(m·s-1)

Solids x-velocity, $$u_{s}$$

5.0

(m·s-1)

Solids y-velocity, $$v_{s}$$

5.0

(m·s-1)

Solids z-velocity, $$w_{s}$$

5.0

(m·s-1)

Fluid temperature, $$T_{g}$$

350.0

(K)

Solids temperature, $$T_{s}$$

300.0

(K)

Gas volume fraction, $$\varepsilon_{g}$$

0.7

Boundary Conditions

All boundaries

Mass inflow

Material properties selected to ensure comparable contribution from convection and diffusion terms. Specified values are constant to avoid the introduction of constitutive laws.

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

## 2.6.3. Results¶

Numerical solutions were obtained using the Central discretization scheme for 8x8, 16x16, 32x32, 64x64, and 128x128 grid meshes. The observed order approaches second order for both $$L_{2}$$ and $$L_{\infty}$$ norms using the Central scheme, as shown in Fig. 2.13. This indicates that the numerical discretization terms have been implemented correctly for all derivative terms within the gas momentum equations, solid momentum equations, gas pressure correction equation, gas energy equation, and solid energy equation.

Fig. 2.13 Observed orders of accuracy for 3D, two-phase flows (constant volume fraction) using (a) $$\mathbf{L}_{\mathbf{2}}$$ norms, and (b) $$\mathbf{L}_{\mathbf{\infty}}$$ norms of the discretization error.