- CFD, Fluid Flow, FEA, Heat/Mass Transfer

Solver Setting

This section deals with solution controls for solvers including topics like CFL Number, Time-step for Transient Simulations, Psuedo-time Marching, Parallel Computing, Nodes and Cluster, HPC - High Performance Computing, Threading, Partitioning, MPI - Message Passing Interface and Scalability.

Any program does only one thing: "Solves exactly what one tells it to solve and how to solve, nothing else!"

Note the the field variables solved during CFD simulation are velocity vector, pressure, temperature (enthalpy). The density at each cell is calculated from equation of state (ideal gas law or real gas laws like Redlich-Kwong and IAPWS equation of state for steam). Note that the "compressible flows" and "compressible gas" are not the same concepts. Compressible flows are characterized by Mach number of the flow whereas compressible behaviour of fluid refers to equation of state such as ideal gas law, essential change in density due to change in pressure and temperature.

Compatibility of several ANSYS FLUENT model with underlying physics and turbulence model are tabulate below (reference ANSYS FLUENT User's Guide)

One of the key consideration is the compressible or incompressible behaviour of the fluid. For liquids under lower pressure conditions (except cases such as hydraulics, deep bore drilling in oil exploration...), they can be assumed incompressible without loss of any accuracy. The compressibility chek for liquids need to be decided based on bulk modulus, operating pressures and coefficient of volume expansion.. The later varies between 0.001 ~ 0.010 [K^{-1}]. The bulk modulus is defined as β = -V × ΔP/ΔV where negative sign refers to the fact that volume decreases as pressure increases. Thus, in terms of density, β = -ρ/Δρ × ΔP. β for water is 2.2 [GPa]. Hence, at pressure of 100 [MPa], the change in water density as compared to density at 1 [atm] would be 1000 [kg/m^{3}] × 0.1 [GPa] / 2.2 [GPa] = 45.5 [kg/m^{3}] which is 4.55% of density of water at 1 [atm].

While the change in density of liquids is more pronouced with change in temperature, the situation is different in cases of gases where density is a strong function of both temperature and pressure changes. Note that the density is calculated based on static conditions (static temperature and static pressure) and not on total conditions (total temperature and total pressure). On the other hand, the stagnation conditions are the true measure of pressure energy and internal energy. There are two modes of reduction in stagnation (or total) pressure: the first mode is due to conversion of pressure energy into velocity (kinetic) energy and the second mode is loss of pressure energy due to wall friction and minor losses (sudden expansion and contraction). The general equations to estimate stagnation and static conditions for all values of flow velocities are described below.

Let's consider isothermal flow through a rough pipe of uniform cross-sectional area. The expression for frictional pressure drop is also described. Here, V denotes area-average velocity at the inlet and the outlet.

From conservation of mass:

Static pressure drop taking into account compressibiity of the fluid:

Since ρ

The operating pressure is used to specify the mean pressure in the system. Though it can be adjusted to convert the gauge pressure into absolute pressure. That is, the operating pressure can be set to zero to get the pressure field directly in absolute values. This is the recommended method for cases involving phase change such as simulation of cavitation in pumps. Note that cavitation occurs when local absolute static pressure in the flow field falls below saturation pressure at given temperature. However, once the operating pressure is set to 0 [Pa], the boundary conditions should also be specified in absolute values, e.g. 1,01,325 [Pa] instead of 0 [Pa].

The solver-setting process encompasses following aspects of numerical solutions.

- Discretization scheme for momentum, pressure, energy and turbulence parameters: The PRESTO (Pressure Staggering Option) is recommended for pressure because in flows with high swirl numbers.
- PV-Coupling such as SIMPLE, SIMPLER, PISO
- Gradient schemes: Green-Gauss node based or Green-Gauss cell-based.
- Conservation target and residual levels for convergence criteria. In addition to the "residual norm", it is strongly recommended to set "mass-conservation" or "energy-conservation" as convergence check.
- Fluid time-scales
- Solid time-scales (in case of conjugate heat transfer or pure conduction problems). The recommended practice is to have solid time-scale set to an order of magnitude (10 times) higher than fluid time-scale.
- One should ensure that the solution is mesh-independent and use mesh adaption to modify the mesh or create additional meshes for the mesh-independence study. A mesh-sensitive result confirms presence of "False Diffusion".
- The node-based averaging scheme is known to be more accurate than the default cell-based scheme for unstructured meshes, most notably for triangular and tetrahedral meshes.
- Note that for coupled solvers which solves psuedo-transient equations even for steady state problems (such as CFX), relaxation factors are not required to be set. The solver applies a false timestep as the convergence process is iterated towards final solution.
- QUICK or Geo-Reconstruct scheme for the volume fraction in FLUENT

SIMPLE Algorithm: Analogy between analytical solution and numerical simulation

- For collocated solvers (such as ANSYS CFX), control volumes are identical for all transport equations, continuity as well as momentum.
- CFX is a node-based solver and constructs control volumes around the nodes from element sectors, thus the number of control volumes is equal to the number of nodes and not the number of elements. The input of CFX may be tetrahedron, prism and hexhedron in physical form, the solver internally generates a polyhedral mesh. In a cell centered code, such as FLUENT or STAR-CD / STAR-CCM+, number of elements are the same as the number of control volumes (as the control volumes are same as physical elements) so these are often used interchangeably.
- All field (or unknown or solution) variables as well as fluid properties are stored at the nodes (the vertices on the mesh).
- The distinction between collocated and staggered approach is nicely explained in section 4.6 by S. V. Patankar in his book "Numerical Heat Transfer and Fluid Flow" and section 7.2 in the book Computational Methods for Fluid Dynamics by J. H. Ferziger and M. Peric.
- For staggered solvers like ANSYS FLUENT, values of scalar field variables as well as material properties are stored as cell centres.
- Due to the difference in the way field variables are stored, the simulation with same mesh, material properties and boundary conditions, the y
^{+}value reported in ANSYS CFX will be approximately twice that reported in FLUENT.

- Smaller physical timesteps are more robust than larger ones.
- An 'isothermal' or cold-flow simulation is more robust than modeling heat transfer. The Thermal Energy model is more robust than the Total Energy Model.
- Velocity or mass specified boundary conditions are more robust than pressure specified boundary conditions. Static pressure boundary is more robust than a total pressure boundary.
- If the characteristic time scale is not simply the advection time of the problem, there may be transient effects holding up convergence. Heat transfer or combustion processes may take a long time to convect through the domain or settle out. There may also be vortices caused by the initial guess, which take longer to move through the entire solution domain. In these cases, a larger timestep may be needed to push things through initially, followed by a smaller timestep to ensure convergence on the small time scale physics. If the large timestep results in solver instability, then a small time scale should be used and more iterations may be required.
- Sometimes the levels of turbulence in the domain can affect convergence. If the level of turbulence is non-physically too low, then the flow might be "too thin" and transient flow effects may be dominating. Conversely if the level of turbulence is non-physically too high then the flow might be "too thick" and cause unrealistic pressure changes in the domain. It is wise to look at the Eddy Viscosity and compare it to the dynamic (molecular) viscosity. Typically the Eddy Viscosity is of the order of 1000 times the dynamic viscosity, for a fully turbulent flow.
- The 2nd Order High Resolution advection scheme has the desirable property of giving 2nd order accurate gradient resolution while keeping solution variables physically bounded. However, may cause convergence problems for some cases due to the nonlinearity of the Beta value. If you are running High Res and are having convergence difficulty, try reducing your timestep. If you still have problems converging, try switching to a Specified Blend Factor of 0.75 and gradually increasing the Blend Factor to as close to 1.0 as possible.

- Re ≤ 5, flow is laminar throughout and no vortices are present downstream of the body.
- 5 < Re < 40, a vortex pair can be observed in the wake region though flow is still laminar in the boundary layer.
- 40 ≤ Re ≤ 200, a von Karman vortex street is formed as where pairs of counter-rotating vortices are shed downstream of the body. The vortex shedding frequency is given by a non-dimensional Strouhal number: St = f × D / U.
- 200 ≤ Re ≤ 300, the flow behind a cylinder undergoes transition to a turbulent wake.
- 300 ≤ Re ≤ 3×10
^{5}, the sub-critical Reynolds number regime wake is turbulent while the boundary layer is still laminar. - Re > 3×10
^{5}, a transition from laminar to turbulent boundary layer separation occurs.

In an explicit Eulerian method, the Courant number cannot be > 1 otherwise it will lead a condition where information is propagating through more than one grid cell at each time step. This is explained in the physical domain of dependence and numerical domain of dependence in following diagram. For a stable solution, numerical domain of dependence must cover the physical (real) domain of dependence.

In multi-dimensional problem, depending upon the spatial and temporal discretization scheme, the CFL condition varies as shown below.

Thus, Courant number can be reduced either by *reducing the time step* or **by increasing the mesh size ** (i.e. coarsening of the mesh). The later option may appear counter-intuitive though mathematically that remains a fact. It is difficult for most of the simulation engineers to believe the fact that coarsening the mesh help them achieve a better convergence.

- The CFL number scales the time-step sizes that are used for the time-marching scheme of the flow solver.
- A higher value leads to faster convergence but can lead to divergence and unstable simulations. The (stiff) CFL condition can be relaxed or softened by using implicit Euler such as Crank–Nicolson which is unconditionally stable.
- The inverse of this is also true where choosing a smaller value in an unstable simulation improves convergence.
- ANSYS FLUENT uses implicit (first order and second order) formulation for pressure-based solver. As per the user guide: "It is best to select the
*Coupled pressure-velocity coupling scheme*if you are using large time steps to solve your transient flow, or if you have a poor quality mesh." - In general, the solution at the end of (inner iterations in) current time-step is used as initial value for iterations at next time-step. ANSYS FLUENT provides option to extrapolate the solution variable values for the next time step using a Taylor series expansion. This can improve the convergence of the transient calculations by enabling and the absolute residual levels are lowered during inner iterations.

To verify that your choice for time-step was proper, check contours of the Courant number within the domain. In FLUENT, select Cell Courant Number under Velocity... For a stable and accurate calculation, the Courant number should not exceed a value of 20-40 in most sensitive transient regions of the domain.

Note that CFX reports two types of Courant number:

The

The value of ΔH^{0} will be negative if the process is endothermic (that is the reaction for material formation requires external heat energy) or positive if the process is exothermic (if heat is released during formation or reaction).

In situations with phase change such as forced convection subcooled nucleate boiling, the standard state enthalpies of vapor and liquid phase are set such that their difference equals to latent heat of vapourization = 2.992E+07 [J/kmole]. The unit of standard state enthalpy is [kJ/kmol] and usually the latent heat value is avaiable in [kJ/kg]. The latest heat value in [kJ/mole] should be multiplied by molecular wight of the liquid. For correct speciffication of latent heat, the reference temperature should be set to 298.15 [K] or 25 [°C]. Note that it is a postive value (in case of boiling simulation) as compared to standard state enthalpies of species such as CO_{2} and CH_{4} (which are negative by sign convection) used in species transport and combustion modeling.

In phase change phenomena (evaporation or condensation), the difference between actual temperature of the phase and saturation conditions are described as degree of subcooling (liquid) and degree of superheating (gas). If the inlet temperature of liquid is T = 350 [K] and the saturation temperature T_{SAT} = 373 [K], liquid state is terms as subcooled and with the degree of subcooling = 373 - 350 = 23 [K]. In a nucleate boiling, when the wall temperature rises above the liquid saturation temperature, steam bubbles will get formed and migrate away from the wall. Since the bulk flow (liquid far away from the wall) is subcooled, these bubbles condense near the center of the pipe.

**Wet Steam**: The steam with droplets of water is known as wet steam and it characterized by dryness fraction. Most of the boilers do not generate dry steam as steam bubbles breaking through the heated surface will pull tiny water droplets along with it. In general, when a substance exists partly as liquid and partly as vapor (gas) phase, its quality is defined as the ratio of the mass of vapor to the total mass. On the hand, mass fraction of the condensed phase is known as wetness factor.

**Dry Steam**: Saturated or superheated steam are known as dry steam.

All the porous media formulation take the form: Δp = -L × (A.v + B.v^{2}) where v is the 'superficial' flow velocity and negative sign refers to the fact that pressure decreases along the flow direction. The 'superficial velocity' is calculated assuming there is no blockage of the flow. L is the thickness of the porous domain in the direction of the flow. Here, A and B are coefficients of viscous and inertial resistances.

In FLUENT, the equation used is: Δp/L = -(μ/α.v + C_{2}.ρ/2.v^{2}) where α is known as 'permeability' and μ is the dynamic viscosity of fluid flowing through the porous domain. This is a measure of flow resistance and has unit of [m^{2}]. Other unit of measurement is the darcy [1 darcy = 0.987 μm^{2}], named after the French scientist who discovered the phenomenon.

STAR-CCM+ uses the expression Δp/L = -(P_{v}.v + P_{i}.v^{2}) for a porous domain.

Steps to find out viscous and inertial resistances:

- Calculate the pressure drop vs. flow velocity data [Δp-v] from empirical correlations or wind-tunnel test or simplified CFD simulations.
- Divide the pressure drop value with thickness of the porous domain. Let's name it as [Δp'-v curve].
- Calculate the quadratic polynomial curve fit coefficients [A, B] from the curve Δp'-v. Ensure that the intercept to the y-axis is zero.
- In STAR-CCM+ these coefficients 'A' and 'B' can be directly used as P
_{v}and P_{i}which are viscous and inertial resistance coefficients respectively. - Divide 'A' by dynamics viscosity of the fluid to get inverse of permeability that is 1/α to be used in ANSYS FLUENT as "viscous resistance coefficient".
- Divide 'B' by [0.5ρ] where ρ is the density of fluid, to get C
_{2}to be used in ANSYS FLUENT as "inertial resistance coefficient". - The method needs to be repeated for the other 3 directions. If the flow is primarily one-directional, the resistances in other two directions need to be set to a very high value, typically 3 order of magnitude higher.

In case porous domain is not aligned to any coordinate direction, the direction of unit vector along the flow and across the flow directions can be estimated from following Javascript program. Note that empty field is considered as 0.0. There is no check if a text value is specified in the input fields and the calculator will result in an error.

First point - X1: | |

First point - Y1: | |

First point - Z1: | |

Second point - X2: | |

Second point - Y2: | |

Second point - Z2: | |

Temperature Drop across a Porous Domain

If flow through the porous domain is assumed (a) incompressible, one-dimension, steady state and (b) pressure drop varies linearly with the flow direction x:

In case of multi-phase flow, an initial distribution of volume fraction of the constituents is important. The method to set initial conditions in ANSYS FLUENT is called patching which requires regions to be marked as shown below. In OpenFOAM setFields (requires setFieldsDict) utility is used to initialize field variables in various zones.

Sometimes k (designated as Tke in STAR CCM+) and and ε (referred to as Tdr in STAR CCM+) may be as high as few thousand and the results are reasonable. The scale can be reduced by turning off the normalization for k and ε. Normalization takes the very first iteration's values to normalize all subsequent residuals. When solution already start with a good solution or guess value for k and ε, the first value is already very low, and residuals may not reduce further. It then sometimes can happen that the residual increases instead of decreasing even though the absolute (without normalization) value might still be very low.

In ANSYS FLUENT, the denominator to calculate the scaled residual for the continuity equation of a pressure-based solver is the largest absolute value of the continuity residual in the first 5 iterations. When iteration starts with a good solution, the largest residual of first 5 iterations may be low. Hence in subsequent iterations, significant reduction in order of magnitude of the residuals may not be observed even though the convergence can be deemed to be of good accuracy. Note 'scaling' and 'normalization' of residuals are different and interpretation should be different as well.

Both elements and nodes are numbered where elements are described as a set of nodes forming its vertices. The more compact is the arrangement of elements and nodes, lesser will be the memory requirements. Some terms associated with elements and node arrangements in a mesh are as follows.

**Bandwidth**: It is the maximum difference between neighboring cells in a zone i.e. if each cell in the zone is numbered in increasing order sequentially, bandwidth is the maximum differences between these indices.- Excerpt from user manual - FLUENT: Since most of the computational loops are over faces, you would like the two cells in memory cache at the same time to reduce cache and/or disk swapping i.e. you want the cells near each other in memory to reduce the cost of memory access.
- In general, the faces and cells are reordered so that neighboring cells are near each other in the zone and in memory resulting in a more diagonal matrix that is non-zero elements are closer to diagonal. Refer to "banded-matrices" in context with numerical simulations.

Multi-phase flows have wide applications in process, automotive, power generation and metal industries including phenomena like mixing, particle-laden flows, CSTR - Contunuously Stirred Tanks Reactor, Water Gas Shift Reaction (WGSR), fluidized bed, fuel injection in engines, bubble columns, mixer vessels, Lagrangian Particle Tracking (LPT). Some of the general characteristics and categories of multi-phase flow are described below along with setting parameters in ANSYS FLUENT

Multiphase flow regimes are typically grouped into five categories: gas-liquid (which are naturally immiscible) flows and (immiscible) liquid-liquid flows, gas-solid flows, liquid-solid flows, three or more phase flows. As can be seen, the immiscibility is a important criteria. In a multi-phase flow, one of the phase is usually continuous and the other phase(s) are dispersed in it. The adjective "Lagrangian" indicates that it relates to the phenomena of tracking a moving points ("fluid particles") - such as tracking a moving vehicle on a road. On the other hand, the adjective "Eulerian" is used to describe correlations between two fixed points in a fixed frame of reference - such as counting the type of vehicles and their speeds while passing through a fixed point on the road.

Gas-liquid flows are further grouped into many categories depending upon the distribution and shape of gas parcels. Three such types are described below.Bubbly Flow: it represents a flow of discrete gaseous or fluid bubbles in a continuous phase.

Slug Flow: This is characterized by flow of large gas bubbles in liquid. Annular Flow: Here one of the phase if confined to area near the wall forming an annular section. Some other types of flows are particle-laden flow such as air carrying dust particles, slurry flow where particles are transported in a liquid, hydrotransport which describes densely-distributed solid particles in a continuous liquid such as cement concrete mix. Gas assisted mixing of solid such as fluidized-bed and settling tank where particles tend to sediment near the bottom of the tank forming thick sludge are some other examples of multi-phase flows.The two dominant method of multi-phase simulations are listed below.

In the Euler-Euler approach, the different phases are treated mathematically as interpenetrating continua where each phase coexist as continuous function in space and time.- A phase cannot be occupied by the other phases in space, the concept of "phasic volume fraction" is used and assumed to be continuous functions of space and time. Distribution of phases are plotted by volume fraction whereas in DPM (Euler-Lagrange approach) the distribution of phases are plotted by particle track.
- Globally, the sum of this phasic volume fraction equals to one.
- Conservation equations for each phase are derived as a set of equations having similar structure for all phases, analogous of momentum equations.
- These equations are closed by constitutive relations obtained from empirical information.

The primary inputs that a user must provide for DPM caculations are the initial conditions that define the starting positions, velocities, parameters for each particle stream and the physical effects acting on the particle streams. Initial conditions of a particle/droplet stream is specfied by creating an 'injection' and by specifying its attributes. The initial conditions describe the instantaneous conditions of an individual particle which are:

- position (x, y, z coordinates) of the particles or parcels
- velocities (u, v, w) of the particle. For cone injections, velocity magnitudes and spray cone angle are used to define the initial velocities
- diameters of the particle
- temperature of the particle
- mass flow rate of the particle stream that will follow the trajectory of the individual particle/droplet though it is required only for coupled calculations
- additional parameters for special injections such as atomizer models

If the field "Number Of Continuous Phase Iterations Per DPM Iteration" field is set to 0, FLUENT will not perform any discrete phase iterations. Excerpts from ANSYS FLUENT: rule of thumb to follow when setting the parameters above is that if you want the particles to advance through a domain consisting of N mesh cells into the main flow direction, the Step Length Factor × N ≈ Max. Number of Steps.

The particle positions are always computed when particles enter/leave a cell even if a very large length scale is specified. The time step used for integration will be such that the cell is traversed in one step.

Also, if "Number Of Continuous Phase Iterations Per DPM Iteration" field is reduced, the coupling between dispersed and continuous flow increases and vice-versa. For a closer coupling, this value should be set to a value ≤ 5 though no unique upper limit exists. Increasing the under-relaxation factor (setting value closer to 1) for discrete phase improves coupling and reducing the under-relaxation factor (setting value away from 1 and closer to 0) for discrete phase weakens the coupling.

DPM Trajectory Equation

Here subscript 'p' denotes 'particle'. Various empirical correlations of drag laws for viscous forces over solid particles are:

ANSYS FLUENT calculates drag coefficient of non-spherical particle using shape factor defined as [s/S] where 's' is the surface area of a hypothetical sphere having the same volume as the particle and 'S' is the actual surface area of the particle.

As explained above, the underlying physics of DPM is described by ordinary differential equations (ODE) as opposed to the Navier-Stokes equation for continuous flow which is expressed in the form of partial differential equations (PDE). Therefore, DPM can have its own numerical mechanisms and discretization schemes, which can be completely different from other numerics (such as discretization schemes) to solve the governing equations. The

Type of coupling: continuous (carrier) phase and discrete phase

The approach differs in ways injected particles (number, volume fraction, injection speed) are treated or the way particle-particle collisions are treated. For example: one particle can represent either one or a group of particles (sometimes known as parcels). In Discrete Particle Method (DPM), particle particle interaction can be treated in following 3-ways:- no collisions at all (used for dilute suspensions): one-way or two-way coupling, Newton's equation of motion for dispersed phase.
- 3-way coupling - soft-sphere approach (the discrete element method, DEM): the instantaneous inter-particle contact forces due to oblique collisions (the normal, damping and sliding forces) are computed using equivalent mechanical elements springs, dashpots and sliders respectively. This approach requires a very small time step which can be as low as < 1 [μs].
- 3-way coupling - hard-sphere approach: interactions between particles are pair-wise (binary collision), instantaneous, momentum-conserving. Faster than soft-sphere method due to feasibility to use longer time steps.

In ANSYS FLUENT, the DDPM option is available only under Eulerian multi-phase model.

In two-way coupling, dispersed phase modifies carrier phase turbulence where bigger particles usually enhance turbulence and smaller particles damp turbulence. This leads to situation where carrier phase turbulence becomes anisotropic. On the other hand, carrier phase turbulence enhances dispersion (fluctuating flow field) and clustering of the dispersed phase (stagnant flow field). When the flow is turbulent, it imparts a random motion on the particles based in instantaneous components of fluctuating velocity. However, RANS flow solution does not resolve all the small–scale turbulent eddies in the flow. The effect of eddies on particle trajectories is resolved using stochastic tracking. Here, a number as specified by "number of tries", particle streams are released from the same point. Each one is given a random "kick" in each grid cell based on the calculated turbulent intensity. This will indicate how turbulence will modify the trajectories.Type of injection

For typical applications, the computational cost of tracking all of the particles individually is prohibitive. Instead, the approach adopted in DPM and Discrete Element Method (DEM) Collision Model is that like particles are divided into parcels and then the position of each parcel is determined by tracking a single representative particle. Thus, in such calculations, following differences between DPM and DEM are mentioned in FLUENT user manual.- The mass used in the DEM calculations of the collisions is that of the entire parcel, not just that of the single representative particle.
- The radius of the DEM parcel is that of a sphere whose volume is the mass of the entire parcel divided by the particle density.

Group injections requires to set the properties position, velocity, diameter, temperature and and flow rate for the First Point (φ_{1}) and Last Point (φ_{N}) in the group. ANSYS FLUENT assigns a value of φ_{i} to the i-th injection in the group using a linear variation between the first and last values for φ. The specified flow rate is defined per particle stream and can also be interpolated linearly between mass flow rates of the first and last injection points.

In ANSYS FLUENT, particle initial conditions (position, velocity, diameter, temperature and mass flow rate) can also be read from an external file to describe injection distribution. The file has the following form with all of the parameters in SI units. All the parentheses are required, name is optional.

(( x y z u v w diameter temperature mass-flow) name)

Sample files generated during sampling of trajectories (path Reports → Sample → Set-up...) for steady particles can also be used as injection files since they have a similar file format. Note that one line in the file = one parcel, 10 lines in the file = 10 parcels, 100 lines in the file = 100 parcels ... "parcels" = groups of particles which have similar properties (diameter, velocity, temperature) as each particle. Parcels are used as computationally it is impossible to track very large number say millions of particles. The mass-flow of the parcel = mass-flow of the group of particles. Generally the more parcels you use, the more accurate the solution shall be.

Example: particle release from a surface - assuming there are 200 faces in the surface.

- The particle injected is sand with density of 2000 [kg/m
^{3}] and diameter 250 [μm]. Thus, mass of each particle is 8.18 [μg] - Total mass flow rate of sand particles = 0.1 [kg/s].
- Total number of particles = 0.10 [kg/s] / 8.18×10
^{-9}[kg] = 1.22×10^{9}[particles/s] - Thus, the 200 trajectories releases from 200 faces of the surface represents 1.22×10
^{9}actual [particles/s]. - If one defines a range of diameters by specifying "minimum diameter", "maximum diameter" and "number of diameters", the total number of trajectories computed will increase to 200 × "number of diameters".
- If stochastic tracking is switched on to account for Turbulent Dispersion and number of tries specified is 20, now the total number of trajectories computed will be 200 × "number of diameters" × "number of tries".

Number of Particle Streams: This option controls the number of droplet parcels that are introduced into the domain at every time step.

Note the options at top right quadrant of "Discrete Phase Model" setting window: Inject particles at (i)Particle Time Step and (ii)Fluid Flow Time Step. The option (i) requires two entrys (a)Particle Time Step Size in [s] and (b) Number of Time Steps. Option (ii) requires only (a) Particle Time Step Size. To start the particle injection at t = 0, set "Start Time" of inject = 0. In order to keep the particle injection long after the time period of interest, set a large value for the Stop Time, for example 500 [s] which ensures that the injection will essentially never stop.Poincaré plots are made by placing a dot on a given surface every time a particle trajectory hits or crosses that surface.

In ANSYS FLUENT, particles can be tracked and their status to be written to files when they encounter selected surfaces. For unsteady particle tracking, the particle trajectories will be updated and the particle states will be written to the files as they cross the selected planes or boundaries. Erosion and accretion rates can be accumulated for repeated trajectory calculations as described below.Summary information (number, mass, average diameter ...) for particles currently in the computational domain can be reported using the Particle Summary dialog box: Reports → Sample → Set-up... It will print following information (their application is mentioned along with the variable).

Total number of parcels, Total number of particles, Total mass, Maximum RMS distance from injector, Maximum particle diameter, Minimum particle diameter, Overall RR Spread Parameter, Maximum Error in RR fit, Overall RR diameter, Overall mean diameter (D_{10}) - Comparisons / evaporation, Overall mean surface area (D_{20}) - Absorption, Overall mean volume (D_{30}) - Hydrology, Overall surface diameter (D_{21}) - Adsorption, Overall volume diameter (D_{31}) - Evaporation / molecular diffusion, Overall Sauter diameter (D_{32}) - Combustion / mass transfer / efficiency studies and Overall De Brouckere diameter (D_{43}) - Combustion equilibrium.

The GNU OCTAVE script used to generate it is as follows.

% Data - number of points, radius, centre of the circle |

n = 2000; RR = 50; xc = 0; yc = 0; |

% Random angular cocordinates |

q = rand(1, n) * (2*pi); r = rand(1, n) * RR * RR; |

x = xc + 0.98 * sqrt(r) .* cos(q); |

y = yc + 0.98 * sqrt(r) .* sin(q); |

a = linspace(0, 2*pi, 100); |

cx = RR * cos(a); cy = RR * sin(a); |

figure(); hold on; box on; |

plot(x, y, 'o'); % Plot the data points |

plot(cx, cy, '-'); % Plot the boundary |

hold off; |

A Python program to generate the relatively uniform distribution of points based on sprials originating from the centre of the disk is summarized below. Reference: stackoverflow.com/ questions/ 27606079/ method-to-uniformly-randomly-populate-a-disk-with-points-in-python

import math |

import random |

import numpy as np |

import matplotlib.pyplot as plt |

n = 2000 |

R = 50 |

alpha = math.pi * (3 - math.sqrt(5)) #The "golden angle" |

phi = random.random() * 2 * math.pi |

x = np.empty((n,1)) |

y = np.empty((n,1)) |

for k in range(1, n): |

q = k * alpha + phi |

r = math.sqrt(float(k)/n) |

x[k-1] = 0.98 * R * r * math.cos(q) |

y[k-1] = 0.98 * R * r * math.sin(q) |

# for plotting circle line: |

a = np.linspace(0, 2*np.pi, 100) |

cx,cy = R * np.cos(a), R * np.sin(a) |

fg, ax = plt.subplots(1, 1) |

ax.plot(cx, cy,'-', alpha=0.5) #Draw circle boundary |

ax.plot(x, y, '.') #Plot points |

ax.axis('square') |

plt.xlim(-R*1.05, R*1.05) |

plt.ylim(-R*1.05, R*1.05) |

plt.show() |

The recommended option is to independently mesh the injection plane and extract centroid of the triangles. This can be achieved by importing the vertices of the triangular elements and finding their centroid as arithmetic average of the coordinates.

The DDPM is based on concept of granular flows and compressible regime (that is, where the solids volume fraction is less than its maximum allowed value described in image above), a solids pressure (which is similar to thermodynamic pressure) is calculated independently and used for the pressure gradient term in the granular-phase momentum equation. The expression of solid pressure is

p_{s} = [1+2(1 + e) α g_{r} ]α ρ_{p} Θ_{s}

The term "granular temperature" has been borrowed from kinetic theory of gases and is a bit misnomer which does not represent any temperature in thermodynamic sense. It represents particle momentum exchange due to translation and collision leading to definition of kinetic and collisional viscosities respectively. The shear and bulk viscosities arising due to this phenomena is added to the solid stress tensor. The granular temperature for the solids phase is proportional to the kinetic energy of the particles’
random motion mathematically expression as Θ_{s} = 1/3[u'.u'] where u' is the fluctuating solids velocity in the Cartesian coordinate system. This is defined as an ensemble average of the particles’ random velocity within a finite volume and time period having particle number per unit volume as averaging basis.

Kinetic viscosity as per Syamlal et al.

Kinetic viscosity as per Gidaspow et al.

The solids bulk viscosity accounts for the resistance of the granular particles to compression and expansion.

- film is small compared to the radius of curvature of the surface so that the properties do not vary across the thickness of the film and
- films formed are thin enough so that the liquid flow in the film can be considered parallel to the wall, with an assumed quadratic shape.

**stick:**where the droplet impacts the wall with little energy and remains nearly spherical**rebound:**where the drop leaves the surface relatively intact but with changed velocity**spread:**where the drop hits the wall with moderate energy and spreads out into the film and**splash:**where part of the impinging drop joins the film and another part of the drop leaves the wall in several other smaller drops.

FSI simulations are classified under a broader category of "Multi-physics Simulations" and not limited only to structural simulations. Fluid-thermal interaction, Fluid-thermal-electric interaction, Fluid-piezoelectric interaction, Structural-magentic interaction ... are other types of multi-physics simulations.

In addition to the settings described above, following steps are required for a FSI simulation:- Solution update can be done through System Coupling component only.
- System Coupling component overrides few settings in participant solver so that the time duration and time step settings are consistent across all participant solvers.
- The surface of solid boundary in contact with fluids need to be set as "Fluid-Solid Interface"in Transient/Static Structural Model.
- In fluid solver such as FLUENT, the "System Coupling" option needs to be activated under "Dynamic Mesh" tab.
- For FLUENT, all regions of type 'Wall' are shown in System Coupling setup.
- For (Transient/Static) Mechanical, all regions of type "Fluid-Solid Interface" appear in System Coupling component.
- Create solid-fluid coupling: select a fluid (FLUENT) and solid (Mechanical) region pair (use Ctrl key for multiple selection) and select "Create Data Transfer" from right click pop-up menu.
- "Data Transfer" defines the Source (participant solver, deforming region and variable being transfered e.g. force, displacement), Target (participant solver, deforming region and variable being transfered e.g. force, incremental displacement) and Data Transfer Controls (data transfer frequency, under-relaxation factor and convergence).
- Post-processing a FSI result:
- Use 'Result' cell in Transient/Static Structural or Fluid Flow (FLUENT) for solver-specific post-processing
- Connect structural 'Solution' cell directly to FLUENT system 'Results' cell or (c)Add a 'Results' System (ANSYS CFD-Post) for unified post-processing of structural and fluid results.

In order to reduce the simulation run time, parallel processing methods have been developed where the arithmetic involved with matrices are broken into segments and assigned to different processors. Some of the the terms associated with this technology are described below. The image from "Optimising the Parallelisation of OpenFOAM Simulations" by Shannon Keough outlines the layout of various components in a cluster.

**HPC:**High Performance Computing - A generic term used to describe the infrastructure (both hardware and software) for parallel processing.**Cluster:**A collection of workstations or nodes connected with each other with high speed network such as 1 GB/s Ethernet network or InfiniBand.**Node:**Each independent component of a cluster or HPC set-up. Each node has following configurations:**Chassis:**For example - HP Z820**CPU:**For example - Intel XEON-E5 2687W**GPU:**Graphics Processing Unit e.g. NVIDIA**Cores:**For example - 8 cores per CPU @ 3.1GHz**RAM:**For example - 8GB DDR3-1600 Registered ECC memory**Storage:**For example - 1TB SATA HDD plus network file system (NFS) server**Operating System:**For example - CentOS 6.5**Application Program:**For example - CFD Software: OpenFOAM or FLUENT V17.2**Latency:**it refers to dormant capacity or sluggishness of the cluster interconnect. In other workds, it also refers to the inertia of the system. Lesser the latency, faster the response time.

**Hyper-threading:**It is a method which allows each core on the CPU to present itself to the operating system as two cores: one real and one virtual. The operating system can then assign jobs to the virtual cores and these jobs are run when the real core would otherwise be idle (such as during memory read/write) theoretically maximising the utilisation of the CPU.- Excerpts from STAR-CCM+ User Manual: "Generally, for best performance, it is recommended that you turn off any features that artificially increase the number of cores on a processor, such as hyper-threading. In situations where hyper-threading is turned on to benefit other applications, you should generally avoid loading more processes than there are physical cores.

In addition to turning off hyper-threading, it is recommended that you set your BIOS settings to favor performance rather than power saving. Almost all heavily parallelized applications suffer performance problems if the CPU frequencies are spun up and down. In most parallel applications and STAR-CCM+ in particular, when one core is operating at a reduced frequency, all other cores are running at lower frequencies as well.

- Excerpts from STAR-CCM+ User Manual: "Generally, for best performance, it is recommended that you turn off any features that artificially increase the number of cores on a processor, such as hyper-threading. In situations where hyper-threading is turned on to benefit other applications, you should generally avoid loading more processes than there are physical cores.
**Memory Intensive Applications:**This refers to the programs whose performance is more influenced by RAM than the clock-speed of the CPU. For example - OpenFOAM**Partitioninng:**This is nothing but a "work distribution" among CPUs. However, tracking of nodes and elements are important. Excerpts from FLUENT user's manual: "Balancing the partitions (equalizing the number of cells) ensures that each processor has an equal load and that the partitions will be ready to communicate at about the same time. Since communication between partitions can be a relatively time-consuming process, minimizing the number of interfaces can reduce the time associated with this data interchange."**MPI:**Message Passing Interface - Each compute node is connected to every other compute node though the network and relies on inter-process communication to perform functions such as sending and receiving arrays, summations over all cells, assembling global matrix - remember A.x = b. Inter-process communication is managed by a message-passing library by an appropriate implementation of the Message Passing Interface (MPI) standard such as OpenMPI, Intel MPI, Cary MPI, IBM Platform MPI, Microsoft MPI, MPICH2 or vendor's own version of MPU.**Scalability:**It refers to decrease in turnaround time for solutions as the number of compute nodes increases. However, beyond a certain point the ratio of network communication time (refer to definition of Cluster above) to computational time increases, leading to reduced parallel efficiency, so optimal system sizing is important for any simulations - determined from a ratio of the time to compute and the time that is taken to exchange data between cluster nodes.**Bandwidth of overall cluster interconnect:**In ANSYS FLUENT, a table containing information about the amount of data communicated within one second between two nodes can be printed to the console. The table summarizes the minimum and maximum bandwidth between two nodes. Checking the bandwidth is particularly useful when you cannot see good scalability with relatively large cases.

Graphics refers to the pictures, videos, charts and animations. The graphics components of a computer control and enhance how these 'graphics' are displayed on computer screen. Usually the graphics components are on a separate card that is plugged into a slot on the motherboard though these can and are built directly into the motherboard as well. Some other names used for the graphics components are video card, video adapter nd *display adapter*. NVIDIA Tesla GPUs for servers and workstations, Quadro GPUs for workstations and GeForce GPUs - gaming class cards are few examples of GPU.

Similar to an operating system which deals with how a computer starts and shuts down, the graphics driver is a program that enables computer to use the graphics card and communicate with CPU, HDD and other components. A 'driver' in general is a software component that lets the operating system and a 'device' communicate with each other. E.g. there is USD 'driver' to make a USB mouse work. The graphics card sends the signal to the computer display and makes the data visible.

Note that computer 'monitors' are not same as "graphics cards or graphics drivers". Graphics card is a hardware and graphics driver is software to make it work. AMD Radeon a graphics card used in my laptop manufactured by Lenovo.

Starting FLUENT in batch mode in **Linux**

fluent | 3d | -g | -i | batch.jou | -tN | -driver | opengl | -rx | 19.2 | -mpi=hp | -cnf=hosts.txt | >& | batch.out |

2d | -gr | -t8 | x11 | 19.1 | intel | ||||||||

2ddp | -gu | -t32 | null | 19.0 | ms | ||||||||

3ddp | -t96 | mpich2 |

Starting FLUENT in batch mode in **Windows**

fluent | 3d | -g | -i | batch.jou | -tN | -driver | opengl | -rx | 19.2 | -mpi=hp | -cnf=hosts.txt | -wait | |

2d | -gr | -t8 | x11 | 19.1 | intel | -hidden | |||||||

2ddp | -gu | -t32 | null | 19.0 | ms | ||||||||

3ddp | -t96 | mpich2 |

- -g indicates that the program (Cortex) is to be run without the GUI and graphics (Linux) or without graphics and with GUI minimized in the taskbar (Windows).
- -gr will run Cortex without graphics. This option can be used in conjunction with the -i option to run a job in background mode.
- -gu will run Cortex without the GUI but with the graphics window(s). (On Windows: it will run FLUENT keeping the window minimized that is the GUI will be available)
- -i reads the specified journal file say batch.jou
- -tN where N = number of processors = 2, 3, 4, ... 8, ... 32, ...
- -driver allows to specify the graphics driver to be used in the solver session
- -gu -driver null allow the generation and export of graphics files from FLUENT simulation run in batch mode
- hosts.txt is the file with the names of the computational nodes (or the hosts)
- Fluent 19.0 onwards, the solver does not differentiate between a
__serial mode__and a*parallel mode*. Now the__serial mode__is in fact the general parallel mode with just one parallel task, -t1 option explained above. - For a ANSYS FLUENT session that is already in progress, the graphics window display can be disabled using the TUI command: display → set → rendering-options → driver → null
- For a ANSYS FLUENT session that is already in progress, to re-enable a graphics window display that had been previously disabled, use the TUI command: (close-all-open-windows); display → set → rendering-options → driver → opengl

Reference: https://doku.lrz.de/pages/viewpage.action?pageId=27394531

> WARNING: Rank:0 Machine mpp2-xxxxxxx has 37 % of RAM filled with file buffer caches. This can cause potential performance issues. Please use -cflush flag to flush the cache. (In case of any trouble with that, try the TUI/Scheme command '(flush-cache)'.)In order to clear the buffer cache, add following lines at the beginning the journal file.

(rpsetvar 'cache-flush/target/reduce-by-mb 12288) (flush-cache)The number 12288 [Mb] = 12 x 1024 [kb] is the specification of the amount of system memory which cannot be flushed out due to boot image of Linux OS files in the memory of the diskless compute nodes. This means that on a compute node with 64 Gb, it tries to flush 64 [Gb] - 12.0 [Gb] = 52 [Gb] of memory. The value of 12288 also varies from cluster-to-cluster and may also change over the time.

The performance meter stores the wall clock time elapsed during a computation as well as message-passing statistics. Since the performance meter is always activated, the statistics can be printed after the computation is completed. To view the current statistics, use the Parallel/Timer/Usage menu item or TUI: paralle/timer/usage. There are many parallel statistics, few easy to decipher are:

**Average wall-clock time per iteration:**it describes the average real (wall clock) time per iteration.**Message count per iteration**: it describes the number of messages sent between all processes per iteration. This is important with regard to communication latency especially on high-latency interconnnects. A higher message count indicates that the parallel performance may be more adversely affected if a high-latency interconnect is being used. Ethernet has a higher latency than Myrinet or Infiniband. Thus, a high message count will more adversely affect performance with Ethernet than with Infiniband.**Data transfer per iteration:**this factor describes the amount of data communicated between processors/iteration. This is important with respect to interconnect bandwidth and is usually dependent on the algorithm being used and the problem being solved. This number generally increases with increases in problem size, number of partitions and physics complexity.**LE wall-clock time per iteration:**the factor that describes the time (wall-clock) spent doing linear equation (LE) solvers (i.e. multigrid).**Total wall-clock time:**it describes the total wall-clock time which can be used to gauge the parallel performance (speedup and efficiency) by comparing this quantity to that from the serial analysis An approximation of parallel speedup may be found in the ratio of Total CPU time to Total wall clock time.

Activating GPU in FLUENT runs

GPGPU stands for General Purpose GPU. The available GPU can be check in FLUENT TUI suing scheme command: /parallel/gpgpu. The GPGPU can be turned on for specific equation solver by TUI: /solve /set /amg-option /amg-gpgpu-option. The options are k, ε, temperature and pressure-velocity coupling.( (air fluid (chemical-formula . #f) (density (constant . 1.225) (premixed-combustion 1.225 300)) (specific-heat (constant . 1006.43)) (thermal-conductivity (constant . 0.0242)) (viscosity (constant . 1.7894e-05) (sutherland 1.7894e-05 273.11 110.56) (power-law 1.7894e-05 273.11 0.666)) (molecular-weight (constant . 28.966)) ) (aluminum solid (chemical-formula . al) (density (constant . 2719)) (specific-heat (constant . 871)) (thermal-conductivity (constant . 202.4)) (formation-entropy (constant . 164448.08)) ) )When FLUENT opens, it auto executes the .fluent file in the: home directory of Linux users, C:\ directory of Windows machines. A .fluent file can contain any number of scheme file names to load. Unless full path is specified for each scheme file in .fluent file, FLUENT tries to locate the scheme files from the working directory. Sample .fluent file is described below. #t is to set boolean flags TRUE and #f is to set it FALSE.

(load "plot.scm") (let ((old-rc client-read-case)) (set! client-read-case (lambda args (apply old-rc args) (if (cx-gui?) (begin ;Do the customization here (rpsetvar 'residuals/plot? #t) (rpsetvar 'residuals/settings '( (continuity #t 0 #f 0.001) (x-velocity #t 0 #f 0.001) (y-velocity #t 0 #f 0.001) (z-velocity #t 0 #f 0.001) (energy #t 0 #f 1e-06) (k #t 0 #f 0.001) (epsilon #t 0 #f 0.001) ) ) (rpsetvar 'mom/relax 0.4) (rpsetvar 'pressure/relax 0.5) (rpsetvar 'realizable-epsilon? #t) (cxsetvar 'vector/style "arrow") ) ) ) ) )

- CFL NUMBER: CFL, it is ratio of characteristic flow velocity to the speed with which time-marching solution advances over a mesh.
- Knudsen number: Kn, defined as the ratio of the mean-free path to the characteristic length scale.
- Mach number: M, defined as the ratio of the characteristic velocity to the sound speed
- Reynolds number: Re, defined as the ratio of the product of the characteristic velocity × characteristic length to the kinematic viscosity. This is one of the widely used number required to distinguish laminar and turbulent flow regimes.
- Strouhal number: Sh, defined as the ratio of the mean-free time to the characteristic time scale. This number is used to express vortex-shedding frequency of wake regions behind bluff-bodies.

The content on CFDyna.com is being constantly refined and improvised with on-the-job experience, testing, and training. Examples might be simplified to improve insight into the physics and basic understanding. Linked pages, articles, references, and examples are constantly reviewed to reduce errors, but we cannot warrant full correctness of all content.

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