1.5 Averaging and closure problems

Averaging and closure problems are crucial concepts in multiphase flow modeling. They involve taking averages of governing equations over many realizations to describe mean flow behavior. This process introduces unclosed terms that require additional modeling to solve the averaged equations.

Ensemble averaging, including Reynolds and Favre methods, is applied to conservation equations. The resulting averaged equations contain unclosed terms like Reynolds stress tensor and turbulent heat flux. Closure models, such as turbulence models and interfacial transfer terms, are needed to express these unclosed terms using mean flow variables.

Ensemble averaging

• Fundamental concept in multiphase flow modeling involves taking averages of the governing equations over a large number of realizations or ensembles
• Ensemble averaging allows for the derivation of averaged conservation equations that describe the mean behavior of the multiphase flow
• Two main types of ensemble averaging commonly used in multiphase flows are Reynolds averaging and Favre averaging

Reynolds vs Favre averaging

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• Reynolds averaging is a conventional method where the flow variables are decomposed into mean and fluctuating components
  • Suitable for incompressible flows or flows with minimal density variations
• Favre averaging, also known as density-weighted averaging, accounts for density fluctuations by introducing a density-weighted average
  • Preferred for compressible flows or flows with significant density variations
• Choice between Reynolds and Favre averaging depends on the nature of the multiphase flow and the assumptions made in the modeling process

Averaging rules and properties

• Ensemble averaging follows certain mathematical rules and properties that simplify the averaging process and the resulting equations
• Linearity property states that the average of a sum is equal to the sum of the averages
  • Allows for the separation and averaging of individual terms in the equations
• Reynolds rules describe the relationships between averages of products and derivatives
  • Product rule: $\overline{ab} = \bar{a}\bar{b} + \overline{a'b'}$
  • Derivative rule: $\overline{\frac{\partial a}{\partial x}} = \frac{\partial \bar{a}}{\partial x}$
• These rules are essential for deriving the averaged conservation equations and handling the closure terms that arise from averaging nonlinear terms

Averaged conservation equations

• Ensemble averaging is applied to the fundamental conservation equations (mass, momentum, energy) to obtain the averaged equations for multiphase flows
• Averaged mass conservation equation:
  • $\frac{\partial \bar{\rho}}{\partial t} + \nabla \cdot (\bar{\rho} \tilde{\mathbf{u}}) = 0$
  • Describes the conservation of mass in terms of averaged density and velocity
• Averaged momentum conservation equation:
  • $\frac{\partial (\bar{\rho} \tilde{\mathbf{u}})}{\partial t} + \nabla \cdot (\bar{\rho} \tilde{\mathbf{u}} \tilde{\mathbf{u}}) = -\nabla \bar{p} + \nabla \cdot \bar{\boldsymbol{\tau}} - \nabla \cdot (\overline{\rho \mathbf{u}' \mathbf{u}'})$
  • Includes terms for averaged pressure, viscous stress, and Reynolds stress tensor
• Averaged energy conservation equation:
  • Similar form to the momentum equation, with additional terms for heat transfer and turbulent heat flux
• These averaged equations form the basis for further modeling and closure of the unclosed terms that arise from the averaging process

Closure problem

• Ensemble averaging of the nonlinear terms in the conservation equations introduces unclosed terms that require additional modeling
• Closure problem refers to the challenge of expressing these unclosed terms in terms of the averaged flow variables
• Unclosed terms arise due to the interaction between fluctuations and the nonlinearity of the governing equations

Unclosed terms in equations

• Reynolds stress tensor: $\overline{\rho \mathbf{u}' \mathbf{u}'}$
  • Represents the transport of momentum due to turbulent fluctuations
• Turbulent heat flux: $\overline{\rho h' \mathbf{u}'}$
  • Represents the transport of heat due to turbulent fluctuations
• Interfacial transfer terms:
  • Momentum transfer: $\overline{\mathbf{M}}_k$
  • Heat transfer: $\overline{Q}_k$
  • Mass transfer: $\overline{\Gamma}_k$
• These unclosed terms require modeling to close the averaged equations and solve for the mean flow variables

Turbulence modeling approaches

• Turbulence modeling aims to provide closure for the unclosed terms related to turbulent fluctuations
• Common approaches include:
  • Eddy viscosity models: Relate the Reynolds stress tensor to the mean velocity gradients through a turbulent viscosity (e.g., mixing length models, k-ε models)
  • Reynolds stress models: Solve transport equations for each component of the Reynolds stress tensor
  • Large Eddy Simulation (LES): Directly resolves large-scale turbulent structures and models the subgrid-scale effects
• Choice of turbulence modeling approach depends on the complexity of the flow, computational resources, and desired level of accuracy

Gradient transport hypothesis

• Commonly used assumption in turbulence modeling that relates the turbulent fluxes to the gradients of mean quantities
• Turbulent viscosity hypothesis:
  • $\overline{\rho \mathbf{u}' \mathbf{u}'} = -\mu_t (\nabla \tilde{\mathbf{u}} + (\nabla \tilde{\mathbf{u}})^T) + \frac{2}{3} \rho k \mathbf{I}$
  • Relates the Reynolds stress tensor to the mean velocity gradients through a turbulent viscosity $\mu_t$
• Turbulent heat flux hypothesis:
  • $\overline{\rho h' \mathbf{u}'} = -\frac{\mu_t}{Pr_t} \nabla \tilde{h}$
  • Relates the turbulent heat flux to the mean enthalpy gradient through a turbulent Prandtl number $Pr_t$
• Gradient transport hypothesis simplifies the closure problem by expressing the unclosed terms in terms of mean quantities and turbulence parameters

Turbulence models for multiphase flows

• Turbulence modeling in multiphase flows requires additional considerations due to the presence of multiple phases and their interactions
• Turbulence models need to account for the effects of phase interfaces, interphase momentum transfer, and turbulence modulation by the dispersed phase
• Common turbulence models used in multiphase flows include mixing length models, two-equation models, and Reynolds stress models

Mixing length models

• Algebraic models that relate the turbulent viscosity to the mean velocity gradients and a characteristic mixing length
• Prandtl mixing length model:
  • $\mu_t = \rho l_m^2 |\nabla \tilde{\mathbf{u}}|$
  • $l_m$ is the mixing length, often prescribed based on the flow geometry or empirical correlations
• Suitable for simple shear flows and boundary layers
• Limited applicability in complex multiphase flows due to the assumption of local equilibrium and the need for case-specific mixing length specifications

Two-equation models

• Widely used turbulence models that solve transport equations for two additional turbulence quantities (e.g., turbulent kinetic energy $k$ and dissipation rate $\varepsilon$)
• Standard k-ε model:
  • Transport equations for $k$ and $\varepsilon$
  • Turbulent viscosity: $\mu_t = C_\mu \rho \frac{k^2}{\varepsilon}$
  • Model constants ($C_\mu$, $\sigma_k$, $\sigma_\varepsilon$, $C_{\varepsilon 1}$, $C_{\varepsilon 2}$) determined from benchmark experiments
• Two-equation models provide a good balance between computational cost and accuracy
• Extensions for multiphase flows include additional terms for interphase turbulence transfer and phase-specific turbulence quantities

Reynolds stress models

• Higher-level turbulence models that solve transport equations for each component of the Reynolds stress tensor
• Avoid the assumption of turbulent viscosity and capture anisotropic turbulence effects
• Reynolds stress transport equations:
  • $\frac{\partial \overline{\rho u_i' u_j'}}{\partial t} + \nabla \cdot (\overline{\rho \mathbf{u} u_i' u_j'}) = P_{ij} + \Pi_{ij} - \varepsilon_{ij} + D_{ij}$
  • Terms for production $P_{ij}$, pressure-strain redistribution $\Pi_{ij}$, dissipation $\varepsilon_{ij}$, and diffusion $D_{ij}$
• Computationally expensive due to the additional transport equations solved
• Suitable for complex multiphase flows with strong anisotropy and significant phase interactions

Interfacial transfer terms

• Multiphase flows involve the exchange of mass, momentum, and energy across phase interfaces
• Interfacial transfer terms appear as source/sink terms in the averaged conservation equations for each phase
• Accurate modeling of interfacial transfer terms is crucial for capturing the coupling between phases and the overall behavior of the multiphase flow

Momentum transfer

• Interfacial momentum transfer includes forces acting on the phase interface, such as drag force, lift force, and virtual mass force
• Drag force:
  • Dominant interfacial force in most multiphase flows
  • Arises from the relative motion between phases and the pressure and shear stress distribution on the interface
  • Modeled using drag coefficients that depend on the flow regime and phase properties
• Lift force:
  • Perpendicular to the relative velocity and arises from the asymmetric pressure distribution around particles or bubbles
  • Important in flows with significant shear or vorticity
• Virtual mass force:
  • Accounts for the acceleration of the surrounding fluid when a particle or bubble accelerates
  • Significant in flows with high-frequency fluctuations or rapid changes in phase velocities

Heat and mass transfer

• Interfacial heat transfer occurs due to temperature differences between phases
  • Modeled using heat transfer coefficients and the temperature difference between the phase interface and the bulk fluid
• Interfacial mass transfer involves the exchange of mass between phases due to phase change processes (e.g., evaporation, condensation) or chemical reactions
  • Modeled using mass transfer coefficients and the concentration difference between the phase interface and the bulk fluid
• Accurate modeling of heat and mass transfer is essential for capturing phase change processes and the overall energy and species balance in the multiphase flow

Closure models for transfer terms

• Closure models are required to express the interfacial transfer terms in terms of the averaged flow variables and phase properties
• Drag force closure:
  • Schiller-Naumann correlation for spherical particles:
    • $C_D = \frac{24}{Re_p} (1 + 0.15 Re_p^{0.687})$ for $Re_p \leq 1000$
    • $C_D = 0.44$ for $Re_p > 1000$
  • Other correlations available for different particle shapes and flow regimes
• Heat transfer closure:
  • Ranz-Marshall correlation for spherical particles:
    • $Nu = 2 + 0.6 Re_p^{0.5} Pr^{0.33}$
  • Correlations based on Nusselt number ($Nu$), Reynolds number ($Re_p$), and Prandtl number ($Pr$)
• Mass transfer closure:
  • Sherwood number correlations similar to heat transfer correlations
  • Depends on the Schmidt number ($Sc$) instead of the Prandtl number
• Selection of appropriate closure models depends on the specific multiphase flow application and the range of operating conditions

Constitutive relations

• Constitutive relations provide additional closure for the averaged conservation equations by relating the stresses and fluxes to the mean flow variables
• In multiphase flows, constitutive relations are required for both the continuous and dispersed phases, as well as for the interfacial interactions
• Common constitutive relations in multiphase flows include interphase drag models, lift and virtual mass forces, and turbulent dispersion force

Interphase drag models

• Interphase drag models describe the momentum transfer between phases due to the relative motion and the pressure and shear stress distribution on the interface
• Drag coefficient models:
  • Schiller-Naumann correlation for spherical particles (as mentioned in the interfacial transfer terms section)
  • Ishii-Zuber correlation for ellipsoidal bubbles or drops
  • Gidaspow drag model for dense particulate flows
• Selection of the appropriate drag model depends on the flow regime (dilute vs. dense), particle shape, and Reynolds number range

Lift and virtual mass forces

• Lift force models account for the transverse force acting on particles or bubbles due to the presence of shear or vorticity in the continuous phase
  • Saffman lift force: $\mathbf{F}_L = C_L \rho_c (\mathbf{u}_c - \mathbf{u}_d) \times (\nabla \times \mathbf{u}_c)$
  • Tomiyama lift force: Includes a correction factor based on the Eötvös number to account for bubble deformation
• Virtual mass force models describe the force experienced by particles or bubbles due to the acceleration of the surrounding fluid
  • Added mass coefficient $C_{VM}$ depends on the particle shape and concentration
  • Virtual mass force: $\mathbf{F}_{VM} = C_{VM} \rho_c (\frac{D\mathbf{u}_c}{Dt} - \frac{d\mathbf{u}_d}{dt})$

Turbulent dispersion force

• Turbulent dispersion force accounts for the dispersion of the dispersed phase due to turbulent fluctuations in the continuous phase
• Arises from the correlation between the fluctuating velocity of the continuous phase and the concentration of the dispersed phase
• Commonly modeled using a gradient diffusion hypothesis:
  • $\mathbf{F}_{TD} = -C_{TD} \rho_c k_c \nabla \alpha_d$
  • $C_{TD}$ is the turbulent dispersion coefficient, often taken as a constant or related to the turbulent Schmidt number
• Turbulent dispersion force is important in flows with significant turbulence and spatial variations in the dispersed phase concentration

Averaging in dispersed flows

• Dispersed flows are a specific type of multiphase flow where one phase is present as discrete elements (particles, droplets, or bubbles) dispersed in a continuous carrier phase
• Averaging approaches in dispersed flows aim to derive macroscopic equations that describe the behavior of the dispersed phase and its interaction with the continuous phase
• Two main averaging approaches used in dispersed flows are the particle-based averaging and the two-fluid model

Particle-based averaging

• Particle-based averaging focuses on the individual dispersed elements and their properties
• Lagrangian approach: Tracks the motion and properties of individual particles or bubbles
  • Suitable for dilute flows with low dispersed phase volume fractions
  • Computationally expensive for large numbers of particles
• Eulerian-Lagrangian approach: Continuous phase is treated as a continuum, while the dispersed phase is tracked individually
  • Coupling between phases through source terms in the continuous phase equations
  • Allows for detailed modeling of particle-fluid interactions and particle collisions

Two-fluid model equations

• Two-fluid model treats both the continuous and dispersed phases as interpenetrating continua
• Derives averaged equations for each phase separately, with coupling terms representing the interfacial interactions
• Averaged equations for the dispersed phase:
  • Mass conservation: $\frac{\partial (\alpha_d \rho_d)}{\partial t} + \nabla \cdot (\alpha_d \rho_d \mathbf{u}_d) = \Gamma_d$
  • Momentum conservation: $\frac{\partial (\alpha_d \rho_d \mathbf{u}_d)}{\partial t} + \nabla \cdot (\alpha_d \rho_d \mathbf{u}_d \mathbf{u}_d) = -\alpha_d \nabla p + \nabla \cdot \boldsymbol{\tau}_d + \alpha_d \rho_d \mathbf{g} + \mathbf{M}_d$
• Interfacial terms ($\Gamma_d$, $\mathbf{M}_d$) represent the mass and momentum transfer between phases
• Two-fluid model requires closure models for the interfacial terms and the dispersed phase stress tensor $\boldsymbol{\tau}_d$

Kinetic theory closure

• Kinetic theory of granular flows provides a framework for modeling the dispersed phase stress tensor and the particle-particle interactions in dense particulate flows
• Treats the dispersed phase as a granular medium and derives constitutive relations based on the kinetic theory of gases
• Key concepts:
  • Granular temperature: Measure of the random fluctuating energy of the particles
  • Solids pressure: Accounts for the particle-particle collisions and the momentum transfer due to the fluctuating motion
  • Granular viscosity: Relates the shear stresses in the dispersed phase to the strain rate and the granular temperature
• Kinetic theory closure models provide constitutive relations for the dispersed phase stress tensor