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.
, including Reynolds and Favre methods, is applied to conservation equations. The resulting averaged equations contain unclosed terms like and . 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 and
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: ab=aˉbˉ+a′b′
Derivative rule: ∂x∂a=∂x∂aˉ
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:
∂t∂ρˉ+∇⋅(ρˉu~)=0
Describes the conservation of mass in terms of averaged density and velocity
Averaged momentum conservation equation:
∂t∂(ρˉu~)+∇⋅(ρˉu~u~)=−∇pˉ+∇⋅τˉ−∇⋅(ρu′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: ρu′u′
Represents the transport of momentum due to turbulent fluctuations
Turbulent heat flux: ρh′u′
Represents the transport of heat due to turbulent fluctuations
Interfacial transfer terms:
Momentum transfer: Mk
Heat transfer: Qk
Mass transfer: Γk
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:
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:
ρu′u′=−μt(∇u~+(∇u~)T)+32ρkI
Relates the Reynolds stress tensor to the mean velocity gradients through a turbulent viscosity μt
Turbulent heat flux hypothesis:
ρh′u′=−Prtμt∇h~
Relates the turbulent heat flux to the mean enthalpy gradient through a turbulent Prandtl number Prt
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:
μt=ρlm2∣∇u~∣
lm 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 ε)
Standard k-ε model:
Transport equations for k and ε
Turbulent viscosity: μt=Cμρεk2
Model constants (Cμ, σk, σε, Cε1, Cε2) 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
Terms for production Pij, pressure-strain redistribution Πij, dissipation εij, and diffusion Dij
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:
CD=Rep24(1+0.15Rep0.687) for Rep≤1000
CD=0.44 for Rep>1000
Other correlations available for different particle shapes and flow regimes
Heat transfer closure:
Ranz-Marshall correlation for spherical particles:
Nu=2+0.6Rep0.5Pr0.33
Correlations based on Nusselt number (Nu), Reynolds number (Rep), 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 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: FL=CLρc(uc−ud)×(∇×uc)
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 CVM depends on the particle shape and concentration
Virtual mass force: FVM=CVMρc(DtDuc−dtdud)
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:
FTD=−CTDρckc∇αd
CTD 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
Interfacial terms (Γd, Md) represent the mass and momentum transfer between phases
Two-fluid model requires closure models for the interfacial terms and the dispersed phase stress tensor τd
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
Key Terms to Review (17)
Closure Approximation: Closure approximation is a method used in multiphase flow modeling to provide a simplified relationship between the moments of the probability distribution function of the phases involved. This concept is crucial when dealing with averaging processes, as it helps to close the system of equations derived from the conservation laws by providing expressions for unclosed terms that arise in the modeling. By making reasonable assumptions about the behavior of the phases, closure approximations enable more tractable mathematical representations of complex multiphase flows.
Closure Relation: A closure relation is a mathematical expression that connects the averages of variables in multiphase flow systems, often used to relate macroscopic and microscopic properties. It serves as a bridge between the averaged quantities and the underlying statistical behavior of the system, allowing for the development of models that can predict system behavior under varying conditions.
Eddy Viscosity: Eddy viscosity is a concept used in fluid dynamics to represent the enhanced mixing and momentum transfer in turbulent flows due to the presence of eddies or swirls. It acts as an effective viscosity that accounts for the effects of turbulence on the flow behavior, allowing for a simplified representation of complex turbulent phenomena. This term is crucial for averaging methods and closure problems, as it helps relate the turbulent quantities to the mean flow characteristics.
Ensemble averaging: Ensemble averaging is a statistical technique used to obtain macroscopic properties of a system by averaging over a large number of microscopic configurations or realizations. This process helps in understanding the behavior of complex systems by providing a bridge between the microscopic and macroscopic views, making it essential in fluid dynamics and multiphase flow analysis. By applying ensemble averaging, we can tackle closure problems that arise when dealing with turbulent flows and other non-linear systems, while also addressing aspects of the continuum hypothesis.
Favre Averaging: Favre averaging is a mathematical technique used in fluid dynamics to calculate averaged properties of a flow field, particularly in the context of multiphase flows. This method distinguishes between the mean flow and fluctuations, allowing for a better representation of non-homogeneous fields by incorporating density variations in the averaging process. It is particularly useful in addressing closure problems that arise from nonlinear terms in governing equations.
Homogenization: Homogenization is a process used in multiphase flow modeling to simplify complex systems by averaging out the properties of the different phases or components. This approach helps in creating a more manageable model that captures the essential behavior of the system without getting lost in intricate details. It plays a crucial role in ensuring that mathematical models remain tractable while still providing meaningful insights into system dynamics and interactions.
Interfacial Tension: Interfacial tension is the force that exists at the interface between two immiscible fluids, which acts to minimize the surface area and create a stable boundary between the fluids. This phenomenon plays a crucial role in various multiphase flow dynamics, affecting how different phases interact, disperse, and behave under various conditions.
Moment Equations: Moment equations are mathematical expressions that describe the balance of forces and torques acting on a fluid system, providing critical insights into the behavior of multiphase flows. These equations account for various forces, such as viscous, pressure, and gravitational forces, and are essential for understanding how different phases interact within a flow. The moment equations play a key role in addressing averaging and closure problems, which arise when dealing with complex fluid systems involving multiple phases or scales.
Navier-Stokes Equations: The Navier-Stokes equations are a set of nonlinear partial differential equations that describe the motion of fluid substances, taking into account viscosity, pressure, and external forces. They are fundamental in modeling fluid flow behavior across various applications, including multiphase flows, by representing how the velocity field of a fluid evolves over time and space.
Phase interaction: Phase interaction refers to the processes that occur between different phases in a multiphase flow system, affecting their behavior and properties. These interactions can significantly influence how the phases mix, transfer momentum and energy, and respond to external forces. Understanding phase interaction is crucial for accurately modeling and predicting the behavior of multiphase systems in various applications.
RANS Equations: RANS equations, or Reynolds-Averaged Navier-Stokes equations, are a set of mathematical equations used to model turbulent fluid flow by averaging the effects of turbulence. They are derived from the Navier-Stokes equations, which describe the motion of fluid substances, and incorporate turbulence models to close the system of equations. This averaging process is essential for capturing the average behavior of fluid flow in complex systems while reducing the computational cost compared to direct numerical simulations.
Reynolds Averaging: Reynolds averaging is a mathematical technique used to analyze turbulent flow by separating the flow variables into a mean component and a fluctuating component. This method is crucial for simplifying the governing equations of fluid dynamics, allowing for a better understanding of turbulence by focusing on averaged quantities rather than instantaneous values. It helps in addressing closure problems that arise when trying to model turbulent flows, making it a fundamental concept in the study of multiphase flow.
Reynolds Stress Tensor: The Reynolds stress tensor is a mathematical representation that quantifies the momentum transfer due to turbulent fluctuations in fluid flow. It arises from the process of averaging the Navier-Stokes equations, where the turbulent velocity components are expressed as deviations from the mean flow. Understanding this tensor is crucial for addressing closure problems in turbulence modeling, as it encapsulates the effects of turbulence on the mean flow properties.
Scaling Laws: Scaling laws are mathematical relationships that describe how physical quantities change when the size or scale of a system is altered. These laws help in understanding the behavior of systems across different scales, making them crucial in modeling multiphase flows and related phenomena. They reveal how various parameters, like velocity or pressure, can vary with size, which is essential for averaging techniques and resolving closure problems in complex flow systems.
Slip Ratio: Slip ratio refers to the relative velocity difference between the phases in a multiphase flow, typically expressed as the ratio of the velocity of one phase to another. This concept is crucial for understanding how different phases, such as gas and liquid, interact and move through a flow system, influencing overall behavior, stability, and phase transitions. Recognizing slip ratios helps in effectively modeling two-fluid interactions and resolving closure problems that arise in multiphase systems.
Turbulent dispersion: Turbulent dispersion refers to the process by which particles or pollutants spread out and mix within a fluid due to the chaotic and fluctuating motions of turbulence. This phenomenon plays a crucial role in how substances are transported and distributed in multiphase flows, affecting both mixing efficiency and concentration gradients. Understanding turbulent dispersion is essential for accurately modeling fluid behavior in complex systems, as it impacts overall flow dynamics and the interactions between different phases.
Turbulent Heat Flux: Turbulent heat flux refers to the transfer of thermal energy due to turbulence in a fluid flow, which plays a crucial role in the heat exchange between different phases or between a fluid and its surrounding environment. This phenomenon is significant in multiphase flow modeling, as it influences temperature distribution, phase interaction, and energy balance within the system. Understanding turbulent heat flux is essential for accurately predicting heat transfer rates and ensuring efficient thermal management in various engineering applications.