Now viewing: EXN/Aero Theory Manual

Jump to User Manual
  1. Introduction
  2. Solver Capabilities
  3. Database Format for Input and Output
  4. EXN/Aero Concept Definitions
    1. Control Volume
    2. Scalar Points
    3. Vector Points
    4. Cell and Interface Design
      1. Cell Objects
      2. Interface Objects
    5. Space-Time Parallelism
  5. Documentation Conventions
  6. Definition of Variables
    1. Reference Density
    2. Density
    3. Reference Temperature
    4. Temperature
    5. Absolute Temperature
    6. Total Temperature
    7. System Pressure
    8. Reference Pressure
    9. Pressure
    10. Absolute Pressure
    11. Dynamic Pressure
    12. Total Pressure
    13. Total Energy
    14. Enthalpy
    15. Total Enthalpy
    16. Volumetric Thermal Expansion
    17. Reynolds Number
    18. Speed of Sound
    19. Mach Number
  7. Basic Equations of Fluid Flow
    1. Conservation Laws and Basic Flow Equations
    2. Continuity Equation
    3. Momentum Equation for Newtonian Fluids
      1. Varying Density
      2. Buoyancy
    4. Energy Equation
    5. General Transport Equation
  8. Equations of State
    1. Thermodynamic Property Models in EXN/Aero
    2. Ideal Gas EOS
      1. Constant specific heat and constant density
      2. Varying Specific Heat
    3. Sutherland's formulas
  9. Turbulence RANS Modeling
    1. Turbulence length scales
    2. Time Averaging - Reynolds Decomposition
    3. Density-Weighted Time Average - Favre Averaging
    4. Flow Equations for Turbulent Incompressible Flow
    5. Flow Equations for Turbulent Compressible Flow
    6. Near Wall Modeling
      1. Inner region
    7. Boussinesq Eddy Viscosity Assumption
    8. Wilcox k-ω Model
    9. Menter Shear Stress Transport k-ω Model
  10. The Large Eddy Simulation (LES) Technique
    1. Spatial Filtering
    2. Filtered Flow Equations
    3. Sub-grid Scale Models for LES
      1. Smagorinsky-Lily model
      2. Canopy K-equation model
      3. WALE model
    4. The Detached Eddy Simulation (DES) Technique
  11. When to use LES\DES
  12. Multiphase Flows
    1. Basic Concepts of Multiphase Flow
    2. Basic Equations
      1. Continuity Equation
      2. Momentum Equation
  13. Initial Conditions
  14. Boundary Conditions
    1. Inlet Boundary Conditions
    2. Outlet Boundary Conditions
    3. No-slip wall
    4. Free-slip wall
    5. Rotating Wall
    6. Moving Wall
    7. Symmetry Plane
  15. Time Discretization
  16. Guidelines for the Efficient Use of EXN/Aero

1.Introduction #

EXN/Aero is a pressure-based, finite volume, fully implicit CFD solver based on the SIMPLEC (Semi-Implicit Method for Pressured Linked Equations – Constistent) method with Algebraic Multigrid (AMG).

The EXN/Aero solver utilizes a hybrid multiblock approach where structured and unstructured mesh regions are concurrently solved in their native data format, preserving the inherent benefits of each type. This allows EXN/Aero to better exploit CPU and GPU resources, playing to the strengths of a heterogeneous compute architecture layout.

EXN/Aero’s true multiblock approach also allows the user to specify the computational precision of each block. In this way, double precision can be applied only where it is needed (e.g. boundary layers), while other areas of the mesh operate in single precision (e.g. far-field mesh). This approach minimizes the memory footprint and increases performance relative to an all-or-nothing double precision approach.

The present manual describes the equations of fluid flow solved by EXN/Aero solver and the concepts for the solution set up.

2.Solver Capabilities #

The EXN/Aero solver provides the following simulation features:

  • LES/DES/RANS.
  • Steady and unsteady turbulent flow calculations.
  • Synthetic turbulence generation for boundary conditions.
  • Dispersed Multiphase.
  • Heat transfer.
  • Compressible flow.
  • Buoyancy driven flows.
  • Calculations on hybrid meshes – structured and unstructured.
  • CGNS database format.
  • Compatibility with 3rd party mesh generators through the CGNS format.
  • Load balancing and resource usage management.
  • Data output in VTK and CGNS format.
  • Porous media.
  • General scalar transport models.
  • Running statistics.

3.Database Format for Input and Output #

EXN/Aero requires as input only mesh data from a single CGNS file. The output data of a simulation, which can be set by the user, are output in VTK format.

 

4.EXN/Aero Concept Definitions #

The EXN/Aero solver relies upon certain definitions used throughout this manual and will be stated here.

4.1.Control Volume #

The control volume (CV) refers to each element employed in the CFD mesh. The EXN/Aero solver can work on hybrid three dimensional meshes comprised of hexahedra, tetrahedra, prismatic and pyramid straight sided elements. General NGON elements are supported as well.

4.2.Scalar Points #

Scalar points refer to the centroids of the CVs and this is where the calculation of scalar variables, e.g. temperature, pressure etc. are defined.

4.3.Vector Points #

Vector points refer to the centroids of the CV and this is where the velocity vector is defined.

 

4.4.Cell and Interface Design #

The cell and interface design allows for greater creativity and flexibility in the way a simulation is parallelized. Our patented design subdivides simulation tasks into two types of objects: cells and interfaces. Subdividing by task and data type is the key for the optimal use of manycore resources, because we have multiple devices, and not all architectures are equally good at every computing task.

Bulky, expensive cell tasks are computed on GPU devices where the thousands of simple processors and fast memory accelerate physics calculations.

Relatively smaller interface tasks are sent to multi-CPU devices, which are suited for complex data transformations and cell-to-cell communication.

Several instances of a simulation can run simultaneously on a single hybrid CPU/GPU resource, which speeds up statistical convergence for unsteady simulations and enables both time and space parallelization.

4.4.1.Cell Objects #

Cell objects (cells) contain the CFD mesh and associated solution data, and represent the bulk of the computational load. Cells might contain structured or unstructured mesh and use single or double precision.

4.4.2.Interface Objects #

Interface objects (interfaces) are multi-function data transformation utilities with many uses, including boundary condition input, cell-to-cell communication, reduction of output data, and interpretation of real-time sensor inputs.

In the schematic below, cells (grey mesh blocks) are connected to each other and the outside world via interfaces (red lines).

 

4.4.5.Space-Time Parallelism #

The EXN/Aero space-time parallelism approach divides the domain in both space and time. Several instances of a spatially-parallelized domain are started simultaneously on the same CPU/GPU machine, where each instance is a “time slab” of the total simulation time window. Time slabs are computed concurrently, and communicate with each other via a predictor-corrector algorithm.

5.Documentation Conventions #

The following conventions are used throughout the present manual:

  • The origin of the coordinate system and the axes directions are chosen to match the mesh coordinates.
  • A fixed Cartesian coordinate system is considered, relative to which, all the equations describing turbulent compressible and incompressible flows are expressed.
  • All the equations in the present manual will be given in tensor form with the spatial coordinates denoted as x_i . That is:x_1\: \text{corresponds to}\: xx_2\: \text{corresponds to}\: yx_3\: \text{corresponds to}\: z
  • Viscous shear stresses are denoted as \tau_{ij} , where the indices i,j refer to the spatial directions.
  • All the indices in the tensor form of the equations refer to the spatial directions, and they number as 1, 2 and 3, corresponding to the x,y,z directions, respectively.
  • For vector variables bold type fonts are used.
  • The Cartesian velocity components are denoted as u_i . That is:

 

<p>u_1\: \text{corresponds to}\: u</p> <p>u_2\: \text{corresponds to}\: v</p> <p>u_3\: \text{corresponds to}\: w</p>

 

  • All quantities are expressed in SI units.
  • Repeated indices follow the summation convention. For example:

 

<p>\frac{1}{2}\rho u_i u_i = \frac{1}{2}\rho\left(u^2_1+u^2_2+u^2_3\right)</p>

6.Definition of Variables #

In the present chapter definitions relevant to the description of fluid flow will be given. These definitions are used later for the development of concepts and mathematical formulations implemented in the EXN/Aero solver.

6.1.Reference Density #

Reference density, \rho_{ref} \left[\text{kg}/\text{m}^3\right] , is an arbitrary datum set by the user for use in buoyant flow calculations. Note that the hydrostatic pressure gradient is affected by the choice of the reference density.

6.2.Density #

Density \rho \left[\text{kg}/\text{m}^3\right]  is the density of the fluid and is calculated by EXN/Aero.

 

 

6.3.Reference Temperature #

Reference temperature T_{ref} \left[\text{K}\right] is set by the user. The reference temperature is arbitrary and used in thermal calculations. Often a good choice is a far-field temperature or inlet value. Reference values are beneficial in avoiding the accumulation of round-off errors.

6.4.Temperature #

The temperature T  [K] is the fluid (thermodynamic) temperature relative to the reference temperature. EXN/Aero calculates temperature directly for incompressible flow. For compressible and incompressible flow simulations, the solver calculates temperature from the enthalpy of the fluid. Inlet and boundary conditions are set using temperature T  [K] and EXN/Aero output displays T  [K].

6.5.Absolute Temperature #

Absolute temperature T_{abs} is the sum of the calculated temperature T and the reference temperature T_{ref}:

 

<p>T_{abs} = T+T_{ref}</p>

6.6.Total Temperature #

The total temperature of a fluid is the temperature the fluid achieves when it is brought to rest adiabatically. Adiabatic conditions exist in the absence of heat transfer.

6.7.System Pressure #

The system pressure is used in the calculation of ideal gas density in incompressible flows when the energy equation is activated. This includes also buoyancy driven flows, but only when local variations in density are used in determining the buoyancy forces. The system pressure should be chosen to represent and average pressure in the domain.

6.8.Reference Pressure #

Reference pressure p_{ref} [Pa] is the pressure level chosen as an arbitrary datum for compressible flow calculations. The value of   p_{ref} is set by the user. Good choices for a reference pressure can be far-field, inlet or outlet pressure specifications. If there are multiple inlets or outlets then an average pressure should be used.  The use of reference conditions helps to minimize the accumulation of round-off errors. For incompressible flow,   p_{ref}  is not specified since the density is independent of the fluid pressure.

6.9.Pressure #

The pressure p [Pa] is the pressure of the fluid (thermodynamic pressure) measured relative to p_{ref} in compressible flows.

6.10.Absolute Pressure #

Absolute pressure p_{abs} is the pressure referenced relative to perfect vacuum and it is equal to the sum of the reference pressure p_{ref} and the pressure p :

<p>p_{abs} = p +p_{ref}</p>

6.11.Dynamic Pressure #

Dynamic pressure p_{d} is defined by the following formula:

<p>p_{d} = \frac{1}{2}\rho u_i u_i</p>

6.12.Total Pressure #

Total pressure p_{tot} is the sum of the absolute pressure and the dynamic pressure:

<p>p_{tot} = p_{abs} + p_d</p>

6.13.Total Energy #

The total energy per unit mass  E \left[\text{J}/\text{kg}\right] of a fluid particle in motion is defined to be the sum of its internal and kinetic energy per unit mass:

<p>E = e+\frac{1}{2}\:u_i u_i</p>

6.14.Enthalpy #

Enthalpy per unit mass h is a thermodynamic quantity which  is defined to be equal to the sum of the internal energy per unit mass e  and the flow work calculated as the absolute pressure p_{abs} divided by density \rho :

 

<p>h = e+\frac{p_{abs}}{\rho}</p>

6.15.Total Enthalpy #

The total enthalpy per unit mass h_{tot} is related to the enthalpy h as follows:

 

<p>h_{tot} = h+\frac{1}{2} u_i u_i</p>

6.16.Volumetric Thermal Expansion #

The volumetric thermal expansion \beta is a thermodynamic quantity defined as follows:

 

<p>\beta = \frac{1}{\rho}\left(\frac{\partial \rho}{\partial T}\right)_p</p>

6.17.Reynolds Number #

The Reynolds number is a measure of the relative importance of inertial and viscous forces. It is defined as follows:

 

<p>Re = \frac{\rho UL}{\mu}</p>

 

where L is a characteristic length of the flow, \mu is the dynamic viscosity of the fluid and U is the flow speed.

6.18.Speed of Sound #

The speed of sound c is the speed at which a isentropic pressure wave propagates in a medium. For an isentropic process, c is defined as follows:

 

<p>c = \sqrt{\left(\frac{\partial p}{\partial \rho}\right)_s}</p>

 

For an ideal gas:

 

<p>c = \sqrt{\gamma\left(\frac{p}{\rho}\right)}</p>

 

where \gamma is the ratio of the specific heats ( c_{p}/c_{V}).

6.19.Mach Number #

The Mach number is the ratio of the speed of the fluid to the speed of sound in the fluid. It is defined as follows:

 

<p>M = \frac{U}{c}</p>

 

where c is the speed of sound and U is the flow speed.

7.Basic Equations of Fluid Flow #

EXN/Aero solves three dimensional unsteady and steady flows for compressible and incompressible fluids. In the present chapter we will present the governing flow equations for each case in conservation form.

 

7.1.Conservation Laws and Basic Flow Equations #

The equations describing fluid flow are mathematical statements of the following physical principles and Newton’s second law of motion:

  • The mass of the fluid is conserved.
  • The rate of change of momentum equals the sum of the forces acting on a fluid particle (Newton’s second law).
  • The rate of change of energy is equal to the sum of the rate of heat addition and the rate of work done on a fluid particle (first law of thermodynamics).

We will not present the derivation of the equations, rather we will give, directly, the mathematical form of the equations and we keep in mind that they express a balance of quantities. Note that repeated indices imply summation.

7.2.Continuity Equation #

The continuity equation expresses the conservation of mass and for a compressible fluid it is expressed as follows:

 

<p>\frac{\partial \rho}{\partial t}+\frac{\partial(\rho u_j)}{\partial x_j} = \frac{D\rho}{Dt}+\rho\frac{\partial u_j}{\partial x_j} = 0\qquad (\text{I})</p>

 

where the total, or substantive, derivative for a flow variable \phi per unit mass is defined as:

 

<p>\frac{D \phi}{Dt} = \frac{\partial \phi}{\partial t} + u_j \frac{\partial \phi }{\partial x_j}\qquad (\text{II})</p>

 

For an incompressible fluid, the derivative of density with respect to time and space is equal to zero, and the continuity equation reduces to:

 

<p>\frac{\partial u_j}{\partial x_j} = 0\qquad (\text{III})</p>

 

7.3.Momentum Equation for Newtonian Fluids #

By virtue of the conservation of mass, the total derivative for a conserved flow variable per unit mass \phi is related to the rate of change per unit volume as follows:

 

<p>\rho\frac{D\phi}{Dt} = \frac{\partial (\rho \phi)}{\partial t}+\frac{\partial {(\rho\phi u_j)}}{\partial x_j}</p>

 

Newton’s second law states that the rate of change of momentum of a fluid particle is equal to the sum of the forces acting on the particle. Two types of forces are present: surface forces and body forces. The contributions of the surface forces (pressure p and viscous stresses \tau_{ij} ) are explicitly stated in the momentum equation, and the contribution of the body forces is represented by a momentum source term S^M_i :

 

<p>\rho\frac{Du_i}{Dt} = -\frac{\partial p}{\partial x_i}+\frac{\partial \tau_{ij}}{\partial x_j} +S^M_i\qquad \text{(I)}</p>

 

For a Newtonian fluid, the viscous stresses are proportional to the rates of deformation, and the stress term can be replaced by:

 

<p>\tau_{ij} = \mu\left(\frac{\partial u_i}{\partial x_j}+\frac{\partial u_j}{\partial x_i}-\frac{2}{3}\frac{\partial u_k}{\partial x_k}\delta_{ij}\right)\qquad \text{(II)}</p>

 

In the EXN/Aero solver, the normal stress is assumed to be part of the fluid pressure, and the momentum equation is solved as:

 

<p>\rho\frac{Du_i}{Dt} = -\frac{\partial p}{\partial x_i}+\frac{\partial }{\partial x_j}\left[\mu\left(\frac{\partial u_i}{\partial x_j}+\frac{\partial u_j}{\partial x_i}\right)\right] +S^M_i\qquad \text{(III)}</p>

 

By default, when no models are activated (e.g. incompressible, isothermal, laminar flow), density is constant, the source term is equal to zero, and viscosity is equal to the dynamic viscosity of the fluid.

7.3.1.Varying Density #

EXN/Aero uses an implicit pressure-based approach for solving compressible flows, rather than a density-based approach.

When the compressible flow option is activated, Eq. (I) in section 7.2 is used, along with the temperature and pressure of the flow to adjust the density of the fluid at each solver iteration.

7.3.2.Buoyancy #

For flows where buoyancy calculations are needed, the following source term is added in the right hand side of the momentum equation given in Eq. (III) in section 7.3:

 

<p>\textbf{S}^M_b = \left(\rho-\rho_{ref}\right)\textbf{g}\qquad \text{(I)}</p>

 

\textbf{g} is the acceleration of gravity. The pressure in this case is modified as follows:

 

<p>p^{\prime}=p+\rho_{ref}\textbf{g}H</p>

 

in order to include the hydrostatic gradient of the reference pressure and H is the height from the horizontal reference plane. It is noted that the pressure p is calculated by EXN/Aero.

 

For buoyant flows with density variations due to small temperature variations, the Boussinesq model is employed. In this model, the common practice is to employ a constant reference density \rho_{ref} for all terms, except for the buoyancy source term, when a thermal expansivity model is used as well. The buoyancy source term is approximated as follows:

 

<p>\rho-\rho_{ref} = \rho_{ref}\beta\left(T-T_{ref}\right)</p>

7.3.4.Energy Equation #

The total energy per unit mass E of a fluid element is defined as the sum of the internal energy and the kinetic energy per unit mass:

 

<p>E = e + \frac{1}{2}u_i u_i\qquad \text{(I)}</p>

 

The energy equation is derived from the first law of thermodynamics:

 

<p>\rho\frac{DE}{Dt} = -\frac{\partial (p u_j)}{\partial x_j}+\frac{\partial }{\partial x_j}\left[u_i \mu\left(\frac{\partial u_i}{\partial x_j}+\frac{\partial u_j}{\partial x_i}\right)\right] -\frac{\partial q_j}{\partial x_j}+S^E\qquad \text{(II)}</p>

 

S^E is a source of energy per unit volume and time, and q_i represents the heat flux, which is given by Fourier’s law:

 

<p>q_j = -\lambda\frac{\partial T}{\partial x_j}\qquad \text{(III)}</p>

 

where \lambda \left[\text{W}/\text{m}\cdot\text{K}\right] is the thermal conductivity.

 

For a compressible flow, Eq. (II) is used to give an equation for the total enthalpy:

 

<p>\rho\frac{Dh_{tot}}{Dt} = \frac{\partial p}{\partial t}+\frac{\partial }{\partial x_j}\left[u_i \mu\left(\frac{\partial u_i}{\partial x_j}+\frac{\partial u_j}{\partial x_i}\right)\right] -\frac{\partial q_j}{\partial x_j}+S^E\qquad \text{(IV)}</p>

 

which is solved in EXN/Aero for a non-isothermal compressible flow. In the case of the flow of liquids or low Mach number flow of gases, the energy equation is expressed with respect to the static enthalpy h by neglecting also the time derivative of the pressure:

 

<p>\rho\frac{Dh}{Dt} = \frac{\partial }{\partial x_j}\left[u_i \mu\left(\frac{\partial u_i}{\partial x_j}+\frac{\partial u_j}{\partial x_i}\right)\right] -\frac{\partial q_j}{\partial x_j}+S^E\qquad \text{(V)}</p>

 

7.3.5.General Transport Equation #

If we consider a general conservative flow variable \phi , the general form of the governing flow equations can be written as follows:

 

<p>\frac{\partial (\rho\phi)}{\partial t} + \frac{\partial (\rho\phi u_i)}{\partial t} = \frac{\partial }{\partial x_i}\left(\Gamma \frac{\partial \phi}{\partial x_j}\right)+S_{\phi}\qquad \text{(I)}</p>

 

where \Gamma represents the diffusion coefficient and S_{\phi} a source term. Eq. (I) highlights the various transport processes: the rate of change term and the convective term on the left hand side and the diffusive term and the source term respectively on the right hand side. In order to bring out the common features of the flow equations, we hide the terms which are not shared between the equations in the source terms. Finally, we note that Eq. (I) is the basis for the finite volume discretization in the EXN/Aero solver.

8.Equations of State #

For the solution of the equations that describe fluid flow we must establish relations between the thermodynamic variables ( \rho,p,T,e, h representing absolute quantities without the abs as a subscript), and moreover, to relate the transport properties ( \mu, \lambda) to the thermodynamic variables. These relations are known as Equations Of State (EOS). Furthermore, according to the state principal of thermodynamics, the local thermodynamic state of a fluid of fixed chemical composition can be fully described by any two independent thermodynamic variables.

For fluids with varying specific heats, we must provide additional information about how the specific heats variate with temperature and pressure (unless it is considered an ideal gas). These relations are known as constitutive equations.

In EXN/Aero, ideal gas models are provided for the calculation of thermodynamic properties. In the most general case (i.e. a real gas), these EOS have the form:

 

<p>\rho = \rho\left(p,T\right)</p>

 

<p>c_p = c_p\left(p,T\right)</p>

 

<p>h=h\left(p,T\right)</p>

 

Enthalpy changes are used for the calculation of thermodynamic variables such as temperature. The general enthalpy differential is expressed as:

 

<p>dh = \left(\frac{\partial h}{\partial p}\right)_p dT+\left(\frac{\partial h}{\partial p}\right)_T dp\qquad \text{(I)}</p>

 

which can be written:

 

<p>dh=c_pdT+\frac{1}{\rho}\left[1+\frac{T}{\rho}\left(\frac{\partial \rho}{\partial T}\right)_p\right] dp\qquad \text{(II)}</p>

 

The second term in Eq. (II) is zero for ideal gases.

 

8.1.Thermodynamic Property Models in EXN/Aero #

Two thermodynamic property models are provided in EXN/Aero: constant density and constant specific heat, and ideal gas EOS with specific heat dependent on temperature T .

8.2.Ideal Gas EOS #

For an ideal gas, the density is calculated using the ideal gas law:

 

<p>\rho= \frac{p}{R\:T}\qquad \text{(I)}</p>

 

where R is the gas constant and is calculated as:

 

<p>R=\frac{R_0}{m}\qquad \text{(II)}</p>

 

where R_0 is the universal gas constant and m is the molecular weight of the gas.

For a fluid that uses the ideal gas EOS the following holds:

 

<p>dh=c_p(T)dT\qquad \text{(III)}</p>

 

where c_p is only a function of temperature. Note that:

 

<p>c_p-c_{V}=\frac{R_0}{m}\qquad \text{(IV)}</p>

 

where c_V is a the specific heat at constant volume.

8.2.1.Constant specific heat and constant density #

The temperature of an ideal gas, in the case of constant c_p is calculated from:

 

<p>h-h_{ref} = c_p\:(T-T_{ref})\qquad \text{(I)}</p>

 

where the reference temperature T_{ref} is set by the user. h_{ref} is calculated internally by the solver. EXN/Aero calculates directly the total or static enthalpy, for compressible and incompressible flows, respectively. Then, from Eq. (I) temperature T is calculated.

 

8.2.2.Varying Specific Heat #

For the case of varying specific heat, the changes in enthalpy are calculated as follows:

 

<p>h-h_{ref} = \int_{T_{ref}}^{T}c_{p}(T)dT\qquad \text{(I)}</p>

 

In this case a polynomial function for c_{p} must be provided to EXN/Aero, that has the form:

 

<p>c_p(T) = A_1+A_2T+A_3T^2+A_4T^3+A_5T^4\qquad \text{(II)}</p>

 

where the constants A_1,A_2,...,A_5 are set by the user.

 

 

8.2.3.Sutherland's formulas #

For flow simulations where the dynamic viscosity is not constant but depends on temperature, we employ Sutherland’s formula given by:

 

<p>\mu = C_1\frac{T^{3/2}}{T+C_2}\qquad \text{(I)}</p>

 

where C_1 and C_2 are constants depending on the gas.

9.Turbulence RANS Modeling #

In the present chapter the equations that describe turbulent compressible and incompressible flows will be presented. Also, the turbulence models implemented in EXN/Aero will be presented.

9.1.Turbulence length scales #

Turbulent flows consist of eddies whose time, velocity and length scales range over a very wide spectrum. The scales of the most energetic eddies are denoted by l,\:T \text{ and } u^{\prime} . Dimensional analysis can be used to obtain approximations to the ratios of these scales for the small and the large eddies. These ratios can be used to gain insight about the scales of turbulent flow structures, and they are expressed as follows:

 

<p>\text{Length-scale ratio}\qquad \frac{\eta}{l} \approx Re_{l}^{-3/4}</p>

 

<p>\text{Time-scale ratio}\qquad \frac{\tau}{T} \approx Re_{l}^{-1/2}</p>

 

<p>\text{Velocity-scale ratio}\qquad \frac{u}{u^{\prime}} \approx Re_{l}^{-1/4}</p>

 

The smallest scales in the flow are set by the viscosity of the flow and are referred to as the Kolmogorov scales.

9.2.Time Averaging - Reynolds Decomposition #

Turbulence manifests as fluctuations, about a mean value, in measurements of flow variables. The mathematical description of turbulence relies on a decomposition of the flow variables into a time-average, or mean, \overline{\phi} and a fluctuating \phi^{\prime} component with zero mean.

This is known as the Reynolds decomposition, where the mean \overline{\phi} , of a flow variable \phi(t) is defined as follows:

 

<p>\overline{\phi} = \frac{1}{\Delta t}\int_{0}^{\Delta t} \phi(t)dt</p>

 

This definition implies that \overline{\overline{\phi}} = \overline{\phi} , and according to Reynolds decomposition:

 

<p>\phi(t) = \overline{\phi} + \phi^{\prime}</p>

 

with the time-average of the fluctuating part, by definition, being equal to zero:

 

<p>\overline{\phi^{\prime}} = \frac{1}{\Delta t}\int_0^{\Delta t}\phi^{\prime}(t)dt = 0</p>

 

For steady state simulations the mean \overline{\phi} is independent of time and for transient simulations the mean corresponds to the ensemble-average.

The statistical tools used to measure the spread of the fluctuations   \phi^{\prime} about the mean value \overline{\phi} are the variance and root mean square (rms):

 

<p>\overline{\left(\phi^{\prime}\right)^2} = \frac{1}{\Delta t}\int_0^{\Delta t}\left(\phi^{\prime}\right)^2dt</p>

 

<p>\phi_{rms} = \sqrt{\overline{\left(\phi^{\prime}\right)^2}}</p>

 

The turbulence kinetic energy per unit mass k at a given location is defined as follows:

 

<p>k=\frac{1}{2}\overline{u_i^{\prime}u_i^{\prime}}\qquad \text{(I)}</p>

 

The turbulence intensity I is defined to be the average rms velocity divided by a reference mean flow velocity U_{ref} , and is linked to the turbulence kinetic energy k as follows:

 

<p>I = \frac{\sqrt{\frac{2}{3}k}}{U_{ref}}\qquad \text{(II)}</p>

9.3.Density-Weighted Time Average - Favre Averaging #

Favre averaging is used in the formulation of the turbulent compressible flow equations in order to separate turbulent fluctuations from the mean flow. For a compressible flow variable \phi the following decomposition is applied:

 

<p>\phi(t)=\tilde{\phi}+\phi^{\prime \prime}</p>

 

where the mean \tilde{\phi} is density-weighted and by definition the mean of the density-weighted fluctuations \rho \phi^{\prime \prime} is zero:

 

<p>\overline{\rho \phi} = \overline{\rho (\tilde{\phi}+\phi^{\prime \prime})} = \overline{\rho}\tilde{\phi}+ \overline{\rho \phi^{\prime \prime}}</p>

 

from which results:

 

<p>\tilde{\phi} = \frac{\overline{\rho \phi}}{\overline{\rho}}</p>

 

since  \overline{\rho \phi^{\prime \prime}} = 0 .

As in time-averaging, for steady state simulations the mean \tilde{\phi} is independent of time and for transient simulations the mean corresponds to the ensemble-average.

 

9.4.Flow Equations for Turbulent Incompressible Flow #

For the case of incompressible flow, substitution of the Reynolds decomposition of the flow variables in the continuity and momentum equations, results in the time-averaged equations.

For simplicity of the presentation the line over the mean variables will be dropped.

 

<p>\frac{\partial u_i}{\partial x_i} = 0\qquad \text{(I)}</p>

 

<p>\frac{\partial u_i}{\partial t} +\frac{\partial (u_iu_j)}{\partial x_j} = -\frac{1}{\rho}\frac{\partial p}{\partial x_i}+\frac{1}{\rho}\frac{\partial }{\partial x_j}\left[\mu\left(\frac{\partial u_i}{\partial x_j}+\frac{\partial u_j}{\partial x_i}-\overline{u_i^{\prime}u_j^{\prime}}\right)\right]+S_i^M\qquad \text{(II)}</p>

 

The extra terms \overline{u_i^{\prime}u_j^{\prime}} appearing in Eq. (II) correspond to the turbulent stresses and are known as the Reynolds stresses. The above equations are known as the Reynolds averaged Navier-Stokes equations (RANS).

9.5.Flow Equations for Turbulent Compressible Flow #

In compressible flow, the mean density varies and the instantaneous density always exhibits fluctuations. However, these fluctuations will be considered negligible at present. For the formulation of the flow equations for turbulent flow, both the Reynolds decomposition and Favre averaging are employed. Specifically, the velocity and internal energy of the flow will be density-weighted (Favre-averaged) and the Reynolds decomposition will be applied to the density and pressure fields. For an unsteady turbulent compressible flow, the density-weighted averaged form of the mean flow equations is the following:

 

Continuity equation:

<p>\frac{\partial \overline{\rho}}{\partial t}+\frac{\partial}{\partial x_j}\left( \overline{\rho} \tilde {u}_j\right) = 0\qquad \text{(I)}</p>

 

Momentum equation:

 

<p>\overline{\rho}\frac{D\tilde{u_i}}{Dt} = -\frac{\partial \overline{p}}{\partial x_i}+\frac{\partial}{\partial x_j}\left[\mu\left(\frac{\partial \tilde{u}_i}{\partial x_j}+\frac{\partial \tilde{u}_j}{\partial x_i}-\overline{\rho u_i^{\prime \prime}u_j^{\prime \prime}}\right)\right] +S^M_i\qquad \text{(II)}</p>

 

Energy equation:

 

<p>\overline{\rho}\frac{D\tilde{h}_{tot}}{Dt} = \frac{\partial \overline{p}}{\partial t}+\frac{\partial }{\partial x_j}\left(\tilde{q}_j-\rho\overline{u_jh}\right)+\frac{\partial }{\partial x_j}\left[\tilde{u}_i\left(\tilde{\tau}_{ij}-\rho\overline{u_i^{\prime \prime}u_j^{\prime \prime}}\right)\right]+S^h</p>

9.6.Near Wall Modeling #

Experimental observations have shown that the turbulent boundary layer adjacent to a solid wall is composed of two regions: the inner region and the outer region.

The inner region is further divided into three zones: the linear sub-layer, where the viscous stresses dominate the flow adjacent to the surface, the buffer layer where viscous and turbulent stresses are of similar magnitude and the log-layer zone where the turbulent stresses dominate. In the outer region the inertia and streamwise pressure of the flow dominates its development.

9.6.1.Inner region #

At the linear sub-layer region viscosity prevails, and the flow does not depend on free stream conditions. The mean flow velocity depends only on the distance from the wall y , the fluid density \rho , viscosity \mu and the wall shear stress \tau_{w} :

 

<p>U = f(y,\:\rho,\:\mu,\:\tau_{w})</p>

 

Dimensional analysis shows that:

 

<p>u^{+}=\frac{U}{u_{\tau}}=f\left(\frac{\rho\: u_{\tau}\:y}{\mu}\right)\qquad \text{(I)}</p>

 

Eq. (I) is the law of the walls and contains the definitions of two dimensionless groups. The non-dimensional wall distance, y^+ , is defined as follows:

 

<p>y^+ = \frac{\rho u_{\tau}y}{\mu}</p>

 

and the non-dimensional velocity u^+ is defined as:

 

<p>u^+=\frac{U}{u_{\tau}}</p>

 

where u_{\tau} is the friction velocity and is equal to:

 

<p>u_{\tau} = \sqrt{\frac{\tau_w}{\rho}}</p>

 

The linear sub-layer is very thin ( y^{+}<5 ) and the shear stress in this zone can be considered constant and equal to the wall stress \tau_{w} throughout its extent. In this region of the boundary layer the following holds:

 

<p>u^+=y^+</p>

 

Because of the above linear relationship the the fluid layer adjacent to the wall is known as the linear sub-layer.

Outside the viscous sub layer (30 < y^{+} < 500) a region exists where viscous and turbulent effects are both important. The shear stress \tau varies slowly with distance from the wall, and within this inner region it is assumed to be constant and equal to the wall shear stress. One further assumption regarding the mixing length allows us to derive a functional relationship between u^+ and y^+ that is dimensionally correct:

 

<p>u^+=\frac{1}{k}\ln(y^+)+B</p>

 

where k \approx 0.4 is the von Karman’s constant and B \approx 5.5 . These constants are valid for all turbulent flows over smooth walls at high Reynolds numbers.

The turbulent Reynolds number is defined as follows:

 

<p>Re_t = \frac{\rho k^2}{\varepsilon} \sim \frac{\mu_t}{\mu}</p>

 

The turbulence eddy dissipation rate \varepsilon is the rate at which the velocity fluctuations dissipate.

9.6.7.Boussinesq Eddy Viscosity Assumption #

Boussinesq’s assumption states that the momentum transfer caused by turbulent eddies can be modeled using an eddy viscosity model. Specifically, the Reynolds stresses are proportional to the mean rate of deformation:

 

<p>S_{ij} = \frac{1}{2}\left(\frac{\partial u_i}{\partial x_j}+\frac{\partial u_j}{\partial x_i}\right)\qquad (\text{I})</p>

 

and they are computed as:

 

<p>-\overline{u_i^{\prime}u_j^{\prime}} = 2\mu_t \left(S_{ij}-\frac{1}{3}\frac{\partial u_k}{\partial x_k}\delta_{ij}\right) -\frac{2}{3}\rho k \delta_{ij}\qquad \text{(II)}</p>

 

where the scalar \mu_t is the eddy viscosity.

9.6.8.Wilcox k-ω Model #

The Wilcox k-\omega model is defined by two equations: one for the turbulence kinetic energy k and one for the turbulence frequency \omega . The eddy viscosity is given by:

 

<p>\mu_t = \frac{\rho\:k}{\omega}</p>

 

with the turbulence frequency defined as:

 

<p>\omega = \frac{\varepsilon}{k}</p>

 

The Reynolds stresses calculation is based on Boussinesq’s approximation given in Eq. (II) in section 9.6.7.

The transport equations for the turbulence kinetic energy k and turbulence frequency \omega for turbulent flows at high Reynolds numbers are:

 

k -equation:

 

<p>\rho\frac{Dk}{Dt} = \frac{\partial}{\partial x_j}\left[\left(\mu+\frac{\mu_t}{\sigma_k}\right)\frac{\partial k}{\partial x_i}\right]+P_k-\beta^{*}\rho k \omega\qquad \text{(I)}</p>

 

\omega -equation:

 

<p>\rho\frac{D\omega}{Dt} =\frac{\partial}{\partial x_j}\left[\left(\mu+\frac{\mu_t}{\sigma_{\omega}}\right)\frac{\partial \omega}{\partial x_j}\right]+\gamma_1\left[\rho S_{ij}S_{ij}-\frac{2}{3}\rho\omega\frac{\partial u_i}{\partial x_j}\delta_{ij}\right]-\beta_1\rho\omega^2\qquad \text{(II)}</p>

 

where the rate of production of turbulence kinetic energy P_k is:

 

<p>P_k = 2\mu_t S_{ij}S_{ij}-\frac{2}{3}\rho\:k\:\frac{\partial u_k}{\partial x_k}\delta_{ij}</p>

 

In EXN/Aero the turbulence constants appearing in Eqs. (I) and (II) are set as follows:

 

<p>\sigma_{k} = 2.0,\qquad \sigma_{\omega} = 2.0,\qquad \gamma_1 = 0.553,\qquad \beta_1 = 0.075,\qquad \beta^*=0.09</p>

 

The Wilcox k-\omega model is suited for:

  • Internal flow problems.
  • For flows exhibiting strong curvature.
  • Separated flows.
  • Jet flows.

It uses wall functions as well, so it does not have high memory requirements. It appears, though, to be sensitive to the initial conditions.

9.6.9.Menter Shear Stress Transport k-ω Model #

The Menter Shear Stress Transport k-\omega (Menter SST k-\omega) turbulence model is a two equations eddy viscosity model. This model combines the aspects of the k-\varepsilon and k-\omega turbulence models. The Reynolds stress computation and the k -equation are the same as in the Wilcox k-\omega model given in Eq. (II) in section 9.6.7 and Eq. (I) in section 9.6.8, respectively. The transport equation for the turbulence frequency \omega is:

 

\omega -equation:

 

<p>\rho\frac{D\omega}{Dt} =\frac{\partial}{\partial x_j}\left[\left(\mu+\frac{\mu_t}{\sigma_{\omega_1}}\right)\frac{\partial \omega}{\partial x_j}\right]+\gamma_2\left[\rho S_{ij}S_{ij} -\frac{2}{3}\rho\omega\frac{\partial u_i}{\partial x_j}\delta_{ij}\right]-\beta_2\rho\omega^2+2\frac{\rho}{\sigma_{\omega,2}\omega}\frac{\partial k}{\partial x_k}\frac{\partial \omega}{\partial x_k}\qquad \text{(I)}</p>

 

The constants for this model are equal to:

 

<p>\sigma_k=1.0,\quad \sigma_{\omega,1} = 2.0,\quad \sigma_{\omega,2}=1.17,\quad \gamma_2 = 0.44,\quad \beta_2 = 0.083,\quad \beta^{*} =0.09</p>

 

In order to alleviate any numerical instabilities caused by the computed values of the eddy viscosity with the standard k-\epsilon model in the far field and the transformed k-\epsilon model near the wall, blending functions are used to achieve a smooth transition between the two models:

 

<p>\alpha = \alpha_1 F_1 +\alpha_2(1-F_1)</p>

 

The eddy viscosity is limited to give improved performance in flows with adverse pressure gradients and wake regions, and the turbulence kinetic energy production is limited to prevent the build-up of turbulence in stagnation regions. The limiter is as follows:

 

<p>\mu_t=\frac{\alpha_1 \rho k}{\text{max}\left(a_1 \omega,SF_2\right)},\quad S =\sqrt{2S_{ij}S_{ij}}</p>

 

where \alpha_1 is a constant and the blending function F_2 is equal to:

 

<p>F_2 = \tanh \left[ \left[\text{max}\left(\frac{2\sqrt{k}}{\beta^*\omega y},\frac{500\nu}{y^2\omega}\right)\right]^2 \right]

 

and,

 

<p>P_k = \text{min}\left(10\beta^*\rho k \omega,2\mu_t S_{ij}S_{ij}-\frac{2}{3}\rho k \frac{\partial u_i}{\partial x_j}\delta_{ij}\right)</p>

 

The SST k-\omega model has proved to be a very good turbulence model for many industrial applications. However, it does not converge very quickly and requires a good resolution of the near-wall region which increases the memory requirements. Its accuracy though is very good. It exhibits sensitivity to the initial conditions and the usual practice is to employ the k-\omega model to compute the flow filed and use it as initial conditions for the SST k-\omega . This model is best suited for:

  • External aerodynamics applications.
  • Boundary layer with adverse pressure gradients or zero pressure gradients.
  • Free shear layers.

10.The Large Eddy Simulation (LES) Technique #

Turbulent flows contain a wide range of temporal and spatial scales. The large scale motions compared to the small ones are generally more energetic. The Large Eddy Simulation (LES) technique is an approach where the large eddies are solved by the flow equations and for the small scale motion a “sub-grid” scale turbulence model is employed. LES is based on a spatial filtering of the flow equations which wipes out the small spatial scales of turbulence. It requires very fine meshes, small time steps and long time integration to generate meaningful statistical correlations for the fluctuating flow variables.

10.1.Spatial Filtering #

A filtered flow variable \overline{\Phi} is defined as follows:

 

<p>\overline{\Phi} (\mathbf{x}) = \int_{D}{\Phi} (\mathbf{x^{\prime}})G(\textbf{x};\mathbf{x^{\prime}})d\mathbf{x^{\prime}}</p>

 

where D is the fluid domain and G is the filter function that determines the resolved eddies. In the finite volume implementation in EXN/Aero the volume averaged “box” (also known as the “top-hat”) filter is employed, and as such, each CV in the computational mesh defines implicitly the filtered function G :

 

<p>\overline{\Phi} (\mathbf{x}) = \frac{1}{V}\int_{D}{\Phi} (\mathbf{x^{\prime}})d\mathbf{x^{\prime}}</p>

 

where V is the volume of the CV. The unresolved part of the \Phi is defined as:

 

<p>\Phi^{\prime} = \Phi-\overline{\Phi}</p>

 

and it is noted that the filtered fluctuations are generally not zero.

10.2.Filtered Flow Equations #

Filtering of the continuity equation given in Eq. (I) in section 6.2 yields the LES continuity equation:

 

<p>\frac{\partial \overline{\rho}}{\partial t}+\frac{\partial (\overline{\rho}\: \overline{u}_j)}{\partial x_j} = 0\qquad \text{(I)}</p>

 

Filtering of the momentum equation in Eq. (III) in section 6.3 yields the LES momentum equation:

 

<p>\frac{\partial (\overline{\rho}\:\overline{u}_i)}{\partial t}+\frac{\partial }{\partial x_j}(\overline{\rho}\:\overline{u}_i\overline{u}_j) = -\frac{\partial \overline{p}}{\partial x_i}+\frac{\partial}{\partial x_j}\left[\mu\left(\frac{\partial \overline{u}_i}{\partial x_j}+\frac{\partial \overline{u}_j}{\partial x_i}\right)\right]+\frac{\partial \tau_{ij}}{\partial x_j}\qquad \text{(II)}</p>

 

where \tau_{ij} is the subgrid-scale (SGS) stress. It includes the effects of the small scales and is defined as follows:

 

<p>\tau_{ij}=-\overline{\rho u_i u_j}+\overline{\rho}\:\overline{u_i}\:\overline{u_j}</p>

 

The large-scale eddies are resolved directly by the filtered flow equations and the effect of the small scales is taken into account by SGS models. In EXN/Aero we employ an eddy viscosity approach where the SGS stresses \tau_{ij} are related to the large-scale strain rate \overline{S_{ij}} as follows:

 

<p>\tau_{ij} = 2\mu_{sgs}\left(\overline{S_{ij}}\: \overline{S_{ij}}-\frac{1}{3}\frac{\partial \overline{u}_k}{\partial x_k}\delta_{ij}\right)</p>

 

10.3.Sub-grid Scale Models for LES #

In LES the effect of the (SGS stresses must be modeled. Moreover, as the small scale motion tends to be isotropic a relatively simple model suffices for turbulence modeling.

10.3.1.Smagorinsky-Lily model #

The Smagorinsky-Lily model is a zero-equation model that scales the SGS turbulent viscosity on the strain rate tensor of the mean flow. The eddy viscosity is modeled as follows:

 

<p>\mu_{sgs} = \rho(C_s\Delta)^2|\overline{S}|</p>

 

where the constant C_s is set equal to 0.1 and,

 

<p>\Delta = V^{1/3}\quad \text{and}\quad\overline{S} = \sqrt{2\overline{S_{ij}}\:\overline{S_{ij}}}</p>

 

The effective viscosity is finally computed as:

 

<p>\mu_{eff}=\mu+\mu_{sgs}</p>

10.3.2.Canopy K-equation model #

The canopy K -equation model is a one-equation SGS stress model that uses the transported SGS turbulence kinetic energy   k and an assumed length scale to compute the SGS turbulent viscosity. Extra source terms are active in the k -equation, and in the momentum equation to model the dissipative, momentum absorbing effect of a vegetative canopy.

10.3.3.WALE model #

The WALE model, also known as Wall Adapting Local Eddy-Viscosity model. This zero-equation model scales SGS turbulent viscosity on both the strain rate tensor and the vorticity tensor, and is generally known for better stress prediction near walls.

10.3.4.The Detached Eddy Simulation (DES) Technique #

LES is still an expensive technique for the computation of boundary layer flows at high Re numbers. An alternative to LES which keeps the advantage of the LES method in flow regions far from walls and employs a RANS formulation for the near wall regions is the Detached Eddy Simulation (DES) technique.

In EXN/Aero for DES simulations the SST turbulence model is employed and the following length scales are used:

 

Length scale for SST:

 

<p>l_{SST} = \frac{\sqrt{k}}{\beta^{*} \omega}</p>

 

Length scale for DES:

 

<p>l_{DES} = \text{min}\left( l_{SST},C_{DES}\Delta \right)</p>

 

where C_{DES} is a constant.

 

11.When to use LES\DES #

When a very accurate resolution of an unsteady flow field is required it is the best choice to use LES or DES. Especially, unsteady three dimensional flows

with vortex shedding or oscillating shear layers, flows with coherent flow structures, studies for noise generation, small scale processes where turbulence resolution is necessary, and good estimation of loads due to fluid flow and cases where RANS simulation fail are cases where the LES or DES technique is adequate. Both, LES and DES techniques are very expensive computationally and usually require expensive HPC systems and long run times for producing statistically meaningful results.

  • Unsteady three dimensional flows with vortex shedding or oscillating shear layers.
  • Unsteady three dimensional flows flow with coherent flow structures.
  • RANS/URANS formulations fail.
  • Studies for noise generation from flows.
  • A good resolution of turbulence is required for flows with small scale processes.
  • A good estimation of unsteady loads due to fluid flow.

12.Multiphase Flows #

In the present chapter we present the mathematical formulation for the treatment of multiphase flows in EXN/Aero. A multiphase system is defined to be a mixture of the phases of solid, liquid and gas. The basic flow equations are expressed relative to averaged quantities, which are derived based on time and spatial averaging. The mixture model is used for the mathematical description of the multiphase flow field.

12.1.Basic Concepts of Multiphase Flow #

We assume a flowing mixture of N phases where one of the phases is a continuous fluid (liquid or gas). In the following developments we denote the quantities referring to the continuous phase with a subscript c . The dispersed phases can be particles, bubbles or droplets. Each phase of the flowing mixture will be referred to using the subscript n .

We define the volumetric fraction \alpha_n of phase n as follows:

 

<p>\alpha_n = \frac{V_n}{V_m}\qquad \text{(I)}</p>

 

where V_n is the volume occupied by phase n  and V_m is the volume of the mixture. It is easy to deduce that:

 

<p>\sum_{i=1}^N \alpha_n=1</p>

 

The mixture density \rho_m is defined as follows:

 

<p>\rho_m = \sum_{n=1}^N\alpha_n\rho_n\qquad \text{(II)}</p>

 

The components of the mixture velocity u_{m,i} are defined as:

 

<p>u_{m,i}=\frac{1}{\rho_m}\sum_{n=1}^N\alpha_n\rho_n u_{n,i} = \sum_{n=1}^N c_n u_{n,i}\qquad \text{(III)}</p>

 

which express the velocity of the center of mass of the mixture. From the previous definition the mass fraction of phase n is defined as:

 

<p>c_n = \frac{\alpha_n \rho_n}{\rho_m}</p>

 

Drift velocity u_{dn,i} is the velocity of phase n relative to the center of mass of the mixture u_{m,i} :

 

<p>u_{dn,i} = u_{n,i}-u_{m,i}\qquad \text{(IV)}</p>

 

The dispersed or slip velocity is the velocity of the dispersed phase relative to the velocity of the continuous phase, and its components are equal to:

 

<p>u_{cn,i} =u_{n,i}-u_{c,i}\qquad \text{(V)}</p>

 

The volumetric flux of phase  n is defined as the volume flow of the phase per unit area and is denoted as j_{n,i} where the subscript i refers to the spatial direction:

 

<p>j_{n,i} = \alpha_n u_{n,i}</p>

 

The mass flux of phase n is defined as follows:

 

<p>G_{n,i} = \rho_n j_{n,i}</p>

 

where i refers to the spatial directions and the following holds:

 

<p>\sum_{n=1}^N G_{n,i} = 0</p>

 

12.2.Basic Equations #

The basic equations of multiphase flow that are solved in EXN/Aero are based on the formulation of the mixture model. In this approach, the continuity and the momentum equation are written for the ensemble of the continuous and the dispersed phases. Moreover, the particle concentrations are calculated from continuity equations for each dispersed phase, and the momentum equations for the dispersed phases are approximated by algebraic equations.

 

In the present manual the Favre-averaged velocity is employed in the multiphase flow equations.

12.2.1.Continuity Equation #

For each phase n, the continuity equation is as follows:

 

<p>\frac{\partial (\alpha_n \rho_n)}{\partial t} + \frac{\partial (\alpha_n \rho_n u_{n,j})}{\partial x_j}= S_n+\sum_{i=1}^N\Gamma_{in}\qquad \text{(I)}</p>

 

where S_n represents the rate of mass generation of phase n and \Gamma_{in} is the mass flow rate per unit volume from phase i to phase n . Summing over all the phases and applying the condition that the total mass is conserved:

 

<p>\sum_{n=1}^N\sum_{i=1}^N\Gamma_{in} = 0</p>

 

we obtain the continuity equation for the mixture:

 

<p>\frac{\partial \rho_m}{\partial t} + \frac{\partial \left(\rho_m u_{m,j}\right)}{\partial x_j} =0\qquad \text{(II)}</p>

 

12.2.2.Momentum Equation #

For each phase n , the momentum equation is as follows:

 

<p>\frac{\partial}{\partial t} (\alpha_n \rho_n u_{n,i}) + \frac{\partial}{\partial x_j} (\alpha_n \rho_n u_{n,i} u_{n,j}) =-\alpha_n\frac{\partial p_n}{\partial x_j}+\frac{\partial }{\partial x_j}\left[\alpha_n\left(\tau_{n,ij}+\tau_{Tn,ij}\right)\right]+\alpha_n\rho_ng_i+S_{n,i}\qquad \text{(I)}</p>

 

where \tau_{n,ij} is the average viscous stress tensor and:

 

<p>\tau_{Tn,ij} = -\overline { \rho_{In}u_{Fn,i}u_{Fn,i}}</p>

 

where \rho_{In} is the instantaneous density of phase n , and   u_{Fn,i} is the fluctuating component of the instantaneous velocity of phase n , u_{In,i}:

 

<p>u_{Fn,i} = u_{In,i}-u_{n,i}</p>

 

Summing over the phases the momentum equation for the mixture results:

 

<p>\rho_m\frac{Du_{m,i}}{Dt} = -\frac{\partial p_m}{\partial x_j}+\frac{\partial }{\partial x_j}\left(\tau_{m,ij}+\tau_{Tm,ij}\right)+\frac{\partial \tau_{Dm,ij}}{\partial x_j}+\rho_m g_i+S_{m,i}\qquad \text{(II)}</p>

 

The three stress tensors appearing in the previous equation are defined as:

 

<p>\tau_{m,ij} = \sum_{n=1}^N\alpha_n \tau_{n,ij}</p>

 

<p>\tau_{Tm,ij} = -\sum_{n=1}^N\alpha_n \overline{ \rho_{In} u_{Fn,i}u_{Fn,i}}</p>

 

<p>\tau_{TDm,ij} = -\sum_{n=1}^N\alpha_n \rho_nu_{dn,i}u_{dn,i}</p>

13.Initial Conditions #

All CFD problems are defined in terms of initial and boundary conditions. At present, the initial conditions in EXN/Aero are specified relative to a fixed Cartesian coordinate system. The units of the variables are expressed in the SI system.

The initial conditions in EXN/Aero are set for the various transport equations for all mesh blocks that are part of a given cell family.

For a turbulent compressible flow the following initial conditions must be set for the solution of the flow equations:

  • Cartesian velocity components u,v,w in m/s.
  • Pressure p in Pa.
  • Temperature T in K.
  • Turbulence intensity I and turbulent length scale l in m, or alternatively, turbulent kinetic energy k in \text{m}^2/\text{s}^2 and turbulence dissipation rate \varepsilon in 1/s.

The values set for temperature and pressure correspond to gauge values and during initialization the reference values are applied as well. That is, temperature and pressure, at a node, are initialized as follows:

 

<p>T_{ini} = T_{ref}+T\qquad p_{ini} = p_{ref}+p</p>

 

where T and p are set by the user. It is noted that density is calculated by the solver using the ideal gas EOS.

For a turbulent incompressible flow the following initial conditions must be set for the solution of the flow equations:

  • Cartesian velocity components u,v,w in m/s.
  • Pressure p in Pa.
  • Temperature T in K, only if the energy equation is solved.
  • Turbulence intensity I and turbulent length scale l in m, or alternatively, turbulent kinetic energy k in \text{m}^2/\text{s}^2 and turbulence dissipation rate \varepsilon in 1/s.

Density is set as a constant, or can be a function of temperature, where in that case it is calculated by using the ideal-gas EOS with a specified (constant) pressure.

 

14.Boundary Conditions #

The following types of boundary conditions are supported by the EXN/Aero solver:

  • Inlet
  • Outlet
  • No-slip wall
  • Free-slip wall
  • Rotating wall
  • Moving wall
  • Symmetry plane
  • Boundaries with Dirichlet conditions (e.g. temperature, pressure)

In EXN/Aero, for the wall type boundary conditions, the option to calculate the resultant force and moment generated by the flow over the wall, exists.

14.1.Inlet Boundary Conditions #

At an inlet boundary, depending on the flow equations that are solved, certain flow variables must be specified.

Since the SIMPLEC algorithm is employed, for the momentum/pressure equation, the user may specify velocity or pressure. If the velocity is chosen it may be specified as follows:

  • Cartesian velocity components u,\:v,\:w .
  • Flow speed normal to the boundary plane.

If the pressure is chosen this has to be in terms of the total pressure p_{tot} , and moreover, if the flow is supersonic the local Mach number M must be specified as well.

For a non-isothermal flow, the temperature T must be specified, and it is noted that the absolute temperature T_{abs} is used in the calculations. It is noted that the temperature set for the boundary and initial conditions is always set relative to T_{ref} .

For turbulent flows the user must specify one of the following two options:

  • Turbulence kinetic energy k and turbulence dissipation rate \varepsilon
  • Turbulence intensity I and length scale.

Also, there is the option for specifying a turbulent profile at an inlet boundary by employing synthetic turbulence data based on two options:

  • Synthetic generated turbulence.
  • Synthetic eddy model.

For the synthetic generated turbulence option the use must provide the following information:

  • Reynolds stress tensor.
  • Wavenumber Filter Low-High.
  • Frequency Filter Low-High.
  • Turbulent Length Scale.

For the synthetic eddy model option the use must provide the following information:

  • Reynolds stress tensor.
  • Turbulent Length Scale.
  • Mean Velocity Magnitude.
  • Location of the min and max.

14.2.Outlet Boundary Conditions #

In EXN/Aero at an outlet boundary, only data for the momentum/pressure equation must be specified in terms of pressure. Specifically, the pressure p  can be set as:

  • Constant pressure.
  • Average pressure.
  • Using a location in the flow field where pressure is set.

Also, the option to disable locally the inflow that might occur can be set.

14.3.No-slip wall #

For the no-slip wall type boundary condition, the relative to the wall flow velocity components at the integration points are set to zero:

 

<p>\mathbf{u}_{wall} = \mathbf{0}</p>

 

Also, we need to specify if the wall is smooth or rough by setting the roughness value.  A smooth wall is defined with a value of zero for the wall roughness.

The thermal treatment of the wall can be defined as follows:

  • Constant temperature wall:

 

<p>T_{wall} = T_{spec}</p>

 

  • Adiabatic Wall (zero heat flux):

 

<p>\mathbf{q} = \mathbf{0}</p>

 

  • Wall heat transfer coefficient:

 

<p>q_{wall} = \theta_{spec}(T-T_{ref,wall})</p>

 

  •  Specific wall heat flux:

 

<p>\mathbf{q}_{wall} = \mathbf{q}_{spec}</p>

 

14.4.Free-slip wall #

In EXN/Aero, the free-slip wall boundary condition type is applied to the mesh regions that define a tangency constraint for the velocity field:

 

<p>\mathbf{u}\cdot{\mathbf{n}} = 0</p>

 

where \mathbf{n} is the surface normal vector. The same thermal treatment exists for this wall type as in the no-slip wall case. The viscous shear stress at a free-slip wall are set to zero:

 

<p>\tau_{wall} = 0</p>

 

The thermal treatment of the wall is the same as for the no-slip wall case.

14.5.Rotating Wall #

In EXN/Aero the rotating wall boundary condition can be applied to rotating no-slip wall regions of the flow domain. The specification of a rotating wall boundary requires the following:

  • Angular velocity vector.
  • Coordinate system location.
  • Wall roughness.

The thermal treatment of the wall is the same as for the no-slip wall case.

14.6.Moving Wall #

In EXN/Aero the moving wall boundary condition can be applied to moving no-slip wall regions of the flow domain. The specification of the boundary requires the following:

  • Wall velocity vector.
  • Wall roughness.

The wall velocity vector is defined with respect to the global frame of reference. The thermal treatment of the wall is the same as for the no-slip wall case.

14.7.Symmetry Plane #

The symmetry boundary condition type is applied to the mesh regions that define a tangency constraint for the velocity field:

 

<p>\mathbf{u}\cdot{\mathbf{n}} = 0</p>

 

where \mathbf{n} is the surface normal vector. It is noted that a symmetry plane is an adiabatic boundary and viscous shear stresses are not zero:

 

<p>\tau_{sym} \neq 0</p>

 

15.Time Discretization #

In EXN/Aero for unsteady flow calculations two time discretization schemes are provided:

  • First order backward Euler scheme.
  • Second order backward Euler scheme.

Both of this schemes are implicit, and thus unconditionally stable.

16.Guidelines for the Efficient Use of EXN/Aero #

Please adhere to these requirements to ensure a fault-free and efficient run on the EXN/Aero solver.

  • Export in a single mesh file in CGNS format from your mesh generator.
  • Subject to the single file requirement, the mesh can be structured, unstructured, or hybrid.
  • Minimum included angle throughout the mesh should be greater than 15 degrees.
  • Aspect ratios less than 1000 in single precision, or else plan to use double precision.
  • Minimize skewness.
  • Check for negative mesh volumes.
  • Label all boundary conditions.
  • As much as possible, group blocks into block families and boundary patches into BC families.
Suggest Edit