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FLUID DYNAMICSMaster Degree Programme in Physics - UNITSPhysics of the Earth and of the Environment

FABIO ROMANELLIDepartment of Mathematics & Geosciences

University of Triesteromanel@units.it

https://moodle2.units.it/course/view.php?id=5449

ComputationalFluid Dynamics(An Intro to)

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What is the goal for using CFD?

Much of the setup for a CFD problem depends on your goal

Achieve final validation of a prototype Identify feasible designs for a new productOptimize product geometryUnderstand complex physical phenomena

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Considerations in CFD:Model Quality vs. Simulation Time

Five components: Geometry: 2D vs 3D complex vs simple

Mesh: cell count, skewness, structured vs. unstructured

Physics: physical phenomena (e.g. free-surface), turbulence model

Solver: numerical methodology, parameters

Hardware: memory, CPU, # of processors

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What is the goal for using CFD?

Much of the setup for a CFD problem depends on your goal

Achieve final validation of a prototype Highly accurate physics and geometry

Identify feasible designs for a new productSimple physics and geometry

Optimize product geometry Simple physics and highly accurate geometry

Capture complex physical phenomenaHighly accurate physics and simple geometry

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Geometry

Import CAD Model or Create GeometryCAD Programs: Solidworks, Creo, Siemens NX, CATIA, Autodesk, etc. CFD programs usually allow for some simplified geometry creation capability

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Geometry Considerations in CFD

2D or 3D Model Symmetry

SymmetricAxisymmetricNone

Highly detailed or simplified Identify flow domain (interior or exterior of the geometry)

All flow takes place IN a geometry, thus for exterior flows, a “bounding box” geometry must be created to encapsulate the area of interest

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Geometry Considerations in CFD: 2D vs 3D Modelling

Using 2D models (as opposed to 3D) is one of the biggest savers of computational time!

Cell count Degrees of Freedom (no z axis)

2D axisymmetric model captures most of what a 3D model will

Even 2D non-axisymmetric models do quite well if 3D geometry doesn’t vary in z direction

However if geometry has complex 3D shape then 2D models are not an optionEven partial 3D geometry can save time compared to full 3D model

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Mesh the Geometry

Discretize the geometry into smaller and (ideally) identical components

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Mesh Considerations in CFD

Meshing is one of the most crucial steps in CFD analysisIt can be considered as an application in itself, with extensive research, mathematics, and software programs dedicated to it

Meshing can have a significant affect on the accuracy of CFD results and how quickly a solution is reached

Poor mesh quality can lead to incorrect results!

Many meshing programs and tools exist to automatically mesh a geometry based on user specified parameters (type of mesh, mesh size, maximum skew, etc.) and the type of solver to be used

Luckily meshing programs contain many checks and warnings to the user to ensure the mesh is of good enough quality, but user expertise is often required for complex cases

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Define Physics

Define the physics that govern the behavior of your modelThe definition of the physics of the CFD analysis determines many of the equations and mathematical models that will be used when solving.

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Physics Considerations in CFD

Fluid dynamics (flow)Turbulence

Heat TransferParticlesFree surface (Volume of Fluid)CavitationElectromagnetism PlasmaStructural Dynamics

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Define Boundary Conditions (B.C.’s)

Define your boundary conditionsThe boundary conditions not only determine how the boundaries of your model behave, but as they are the only input/output to the model, they affect how the entire model behaves.

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B.C. Considerations in CFD

Proper boundary condition definition is crucial to ensuring accurate CFD modelling.

Poor boundary condition definition can cause unrealistic model behavior and thus incorrect results

Many types of boundary conditionsInlet

Velocity, Pressure, Mass Flow

OutletConservative, Pressure, Mass Flow

WallSlip, non-slip

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Solver Considerations in CFD

Select your solver and define the parametersThe solver parameters determine how the solver behaves and the equations used to interpolate and manipulate the numerical schemes

The solver settings and parameters can have a huge impact on simulation time and the accuracy of results

Many different parameters/settingsType of Solver (i.e. Numerical Methodology)

FVM, FEM, FDM, BEM

Differential SchemesFirst-order upwind, Central differencing, QUICK

Convergence Criteria# of iterations, residual value

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Solve and Post-Process

Click “Solve” or “Start” or “Run” and analyze your convergence criteria.

BEWARE: Convergence of the simulation does not mean that your solution is correct!

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CFD Solution MethodsVarious solution methods for solving the Governing Equations

FDM, FEM, FVM, BEM, Lattice Boltzmann, etc.

Several Commonalities Domain is discretized into series of grid points/control volumes

Discretize the governing equations (i.e. convert to algebraic form)Apply the Boundary ConditionsSolve the equations

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Model Equations

Most commercial CFD codes solve the continuity, Navier-Stokes, and energy equations

Coupled, non-linear, partial differential equationsFor example, incompressible form

∂u∂x

+ ∂v∂y

+ ∂w∂z

= 0

∂u∂t

+ u ∂u∂x

+ v ∂u∂y

+ w ∂u∂z

= − 1ρ∂p∂x

+ µρ

∂2u∂x2

+ ∂2u∂y2

+ ∂2u∂z2

⎛⎝⎜

⎞⎠⎟

∂v∂t

+ u ∂v∂x

+ v ∂ρv∂y

+ w ∂ρv∂z

= − 1ρ∂p∂y

+ µρ

∂2v∂x2

+ ∂2v∂y2

+ ∂2v∂z2

⎛⎝⎜

⎞⎠⎟

∂w∂t

+ u ∂w∂x

+ v ∂w∂y

+ w ∂w∂z

= − 1ρ∂p∂z

+ µρ

∂2w∂x2

+ ∂2w∂y2

+ ∂2w∂z2

⎛⎝⎜

⎞⎠⎟

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Model Equations

Most commercial CFD codes solve the continuity, Navier-Stokes, and energy equations

Coupled, non-linear, partial differential equations

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Discretization - Algebraic equations

To solve NSE, we must convert governing PDE’s to algebraic equations

Finite difference methods (FDM)Each term in NSE approximated using Taylor series, e.g.,

Finite volume methods (FVM)Use CV form of NSE equations on each grid cell.Most popular approach, especially for commercial codes

Finite element methods (FEM)Solve PDE’s by replacing continuous functions by piecewise approximations defined on polygons, which are referred to as elements. Similar to FDM.

∂U∂x

=Ui+1 −Ui

Δx+O Δx( )

∂2U∂x2

=Ui+1 − 2Ui +Ui−1

Δx( )2+O Δx( )2

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Finite Volume Method (FVM)

Most common method in commercial CFD softwareBasic Methodology

Divide the flow domain into control volumesIntegrate the differential equation over the control volume and apply the divergence theoremAssumptions made about how flow property values vary at control volume faces in order to evaluate the derivative termsResults is a set of linear algebraic equations for each control volumeSolve system of equations for each control volume

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Finite Volume Method: the Solution Domain

Solution domain is subdivided into control volumes (cells) by a grid.The grid defines the boundaries of the control volumes, while the computational node is at the center.Flow property values are calculated at the computational node, with values at the faces interpolated from adjacent nodes.

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Finite Volume Method: Control Volume

The net flux through the control volume boundary is the sum of the integrals over the control volume faces

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Finite Volume Method: Discretization Example

We will illustrate how the conservation equations used in CFD can be discretized by examining chemical species transport in the flow field to the right The species transport equation (given incompressible flow) is:

c is concentration of the chemical species, D is the diffusion coefficient, and S is a source term

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Finite Volume Method: Discretization Example

The balance over the control volume (cell p) is given by

However, this equation contains the concentration values at the faces, which need to be interpolated from the cell centers

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Finite Volume Method: Discretization Example

To interpolate the values at the cell faces, we will use the first order upwind scheme. This scheme assumes that the value at the face is equal to the value of the cell center directly upstream from it. Thus the equation transforms from:

To:

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Finite Volume Method: Discretization Example

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Finite Volume Method: Interpolation – Finding Face Values

Fluid properties are always calculated/defined at cell centers, thus assumptions need to be made about the interpolation of that value between cell centers, to cell facesSeveral numerical schemes can be used:

First-order upwindCentral differencingPower-lawSecond-order upwindQUICK

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Finite Volume Method: First Order Upwind Scheme

Value at the cell face is equal to the value at the cell center upstream of that face

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Finite Volume Method: Central Differencing Scheme

Value at the cell face is determined by linear interpolation between the cell centers adjacent to that face

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Finite Volume Method: Power-law scheme

The value at the cell face is determined by an exponential profile through the adjacent cell center values

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Finite Volume Method: Second-order Upwind Scheme

The value of the cell face is determined by linear extrapolation from the two upstream cell centers

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Finite Volume Method: QUICK Scheme

Quadratic Upwind Interpolation for Convective Kinetics

A quadratic curve is fitted through the two upstream and one downstream cell centers

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Finite Volume Method: Reaching a Solution

As mentioned, each of the governing equations is converted to algebraic form and solved for each of the cell centers, this is essentially a “best guess” to the true solution, but still far off from it. With these values, the systems of equations is again solved at each cell to come up with new values, another “best guess” to the true solution and a bit closer. This iterative process is continued until the change in value from one iteration to the next becomes to small that the solution can be considered converged (i.e. the true solution).

Flow field and scalar fields do not change between iterations when converged!

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Finite Volume Method: Residuals

The residuals are one method for judging if the solution is converged

Measure error in conservation equationsGOAL IS TO MINIMIZE RESIDUALS

Absolute residual at cell p is defined as:

Typically this is scaled relative to the local value, and summed across the entire domain to determine:

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Finite Volume Method: Ensuring Convergence

While residuals are one of the most common ways to judge convergence, they are not the end all be all.

Typically residuals need to be on the order of 1E-3 to 1E-4 to consider a solution converged.

But a converged solution may have high residuals while a non-converged solution has residual values in the 1E-3 to 1E-4 range.

Other methods to monitor convergence include calculating forces on a object, monitoring temperature/flow plots to ensure they are not changing between iterations

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Discretization - Grid Generation

Flow field must be treated as a discrete set of points (or volumes) where the governing equations are solved.Many types of grid generation: type is usually related to capability of flow solver.

Structured gridsUnstructured gridsHybrid grids: some portions of flow field are structured (viscous regions) and others are unstructuredOverset (Chimera) grids

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Structured Grids

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Unstructured Grids

Branches in Human LungStructured-Unstructured Nozzle Grid

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Structured Overset Grids

Submarine

Moving Control Surfaces Artificial Heart Chamber

Surface Ship Appendages

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Overset Chimera

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Boundary Conditions

Typical conditionsWall

No-slip (u = v = w = 0)Slip (tangential stress = 0, normal velocity = 0)With specified suction or blowingWith specified temperature or heat flux

InflowOutflowInterface Condition, e.g. Air-water free surfaceSymmetry and Periodicity

Usually set through the use of a graphical user interface (GUI) – click & set

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Example: face and cell zones associated with pipe flow through orifice plate

inlet

outlet

wall

orifice(interior)

orifice_plate and orifice_plate-shadow

fluid

BC Overview

Boundary conditions are a required component of the mathematical model.Boundaries direct motion of flow. Specify fluxes into the computational domain, e.g. mass, momentum, and energy.Fluid and solid regions are represented by cell zones.Material and source terms are assigned to cell zones.Boundaries and internal surfaces are represented by face zones.Boundary data are assigned to face zones.

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Neumann and Dirichlet boundary conditions

When using a Dirichlet boundary condition, one prescribes the value of a variable at the boundary, e.g. u(x) = constant.When using a Neumann boundary condition, one prescribes the gradient normal to the boundary of a variable at the boundary, e.g. ∂nu(x) = constant.When using a mixed boundary condition a function of the form au(x)+b∂nu(x) = constant is applied.Note that at a given boundary, different types of boundary conditions can be used for different variables.

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Flow inlets and outlets

A wide range of boundary conditions types permit the flow to enter and exit the solution domain:

General: pressure inlet, pressure outlet.Incompressible flow: velocity inlet, outflow.Compressible flows: mass flow inlet, pressure far-field.Special: inlet vent, outlet vent, intake fan, exhaust fan.

Boundary data required depends on physical models selected.General guidelines:

Select boundary location and shape such that flow either goes in or out. Not mandatory, but will typically result in better convergence.Should not observe large gradients in direction normal to boundary near inlets and outlets. This indicates an incorrect problem specification.Minimize grid skewness near boundary.

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Pressure boundary conditions require static gauge pressure inputs:

The operating pressure input is set separately.Useful when:

Neither the flow rate nor the velocity are known (e.g. buoyancy-driven flows).A “free” boundary in an external or unconfined flow needs to be defined.

gauge/static pressure

operating pressure

pressure level

operating pressure

absolute pressure

vacuum

Pressure boundary conditions

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One defines the total gauge pressure, temperature, and other scalar quantities at flow inlets:

Here k is the ratio of specific heats (cp/cv) and M is the Mach number. If the inlet flow is supersonic you should also specify the static pressure.Suitable for compressible and incompressible flows. Mass flux through boundary varies depending on interior solution and specified flow direction.The flow direction must be defined and one can get non-physical results if no reasonable direction is specified. Outflow can occur at pressure inlet boundaries. In that case the flow direction is taken from the interior solution.

incompressible flows

compressible flows

Pressure inlet boundary (1)

ptotal = pstatic +12ρV 2

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For non-isothermal incompressible flows, one specifies the inlet temperature.For compressible flows, one specifies the total

temperature T0, which is defined as the temperature that the flow would have if it were brought to a standstill under isentropic conditions:

Here k is the ratio of specific heats (cp/cv), M is the Mach number, and Ts is the static temperature.

Pressure inlet boundary (2)

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Pressure outlet boundary

Here one defines the static/gauge pressure at the outlet boundary. This is interpreted as the static pressure of the environment into which the flow exhausts.Usually the static pressure is assumed to be constant over the outlet. A radial equilibrium pressure distribution option is available for cases where that is not appropriate, e.g. for strongly swirling flows.Backflow can occur at pressure outlet boundaries:

During solution process or as part of solution.Backflow is assumed to be normal to the boundary.Convergence difficulties minimized by realistic values for backflow quantities.Value specified for static pressure used as total pressure wherever backflow occurs.

Pressure outlet must always be used when model is set up with a pressure inlet.

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Defines velocity vector and scalar properties of flow at inlet boundaries.Useful when velocity profile is known at inlet. Uniform profile is default but other profiles can be implemented too.Intended for incompressible flows. The total (stagnation) properties of flow are not fixed. Stagnation properties vary to accommodate prescribed velocity distribution. Using in compressible flows can lead to non-physical results.Avoid placing a velocity inlet too close to a solid obstruction. This can force the solution to be non-physical.

Velocity inlets

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Outflow boundary

Outflow boundary conditions are used to model flow exits where the details of the flow velocity and pressure are not known prior to solution of the flow problem. Appropriate where the exit flow is close to a fully developed condition, as the outflow boundary condition assumes a zero normal gradient for all flow variables except pressure. The solver extrapolates the required information from interior.Furthermore, an overall mass balance correction is applied.

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outflow condition ill-posed

outflow condition not obeyed

outflow condition obeyed

outflow condition closely obeyed

Restrictions on outflow boundaries

Outflow boundaries cannot be used:With compressible flows.

With the pressure inlet boundary condition (use velocity inlet instead) because the combination does not uniquely set a pressure gradient over the whole domain.

In unsteady flows with variable density.

Do not use outflow boundaries where:

Flow enters domain or when backflow occurs (in that case use pressure b.c.).Gradients in flow direction are significant.Conditions downstream of exit plane impact flow in domain.

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Using outflow boundary condition:Mass flow divided equally among all outflow boundaries by default.Flow Rate Weighting (FRW) set to one by default.For uneven flow distribution one can specify the flow rate weighting for each outflow boundary: mi=FRWi/ΣFRWi. The static pressure then varies among the exits to accommodate this flow distribution.

Can also use pressure outlet boundaries to define exits.

pressure-inlet (p0,T0) pressure-outlet (ps)2

velocity-inlet (v,T0)pressure-outlet (ps)1

or

FRW2

velocity inlet

FRW1

Modeling multiple exits

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Other inlet and outlet boundary conditions

Mass flow inlet.Used in compressible flows to prescribe mass flow rate at inlet.Not required for incompressible flows.

Pressure far field.Available when density is calculated from the ideal gas law.Used to model free-stream compressible flow at infinity, with free-stream Mach number and static conditions specified.

Exhaust fan/outlet vent.Model external exhaust fan/outlet vent with specified pressure jump/loss coefficient and ambient (discharge) pressure and temperature.

Inlet vent/intake fan.Model inlet vent/external intake fan with specified loss coefficient/ pressure jump, flow direction, and ambient (inlet) pressure and temperature.

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Determining turbulence parameters

When turbulent flow enters domain at inlet, outlet, or at a far-field boundary, boundary values are required for:

Turbulent kinetic energy k.

Turbulence dissipation rate ε.Four methods available for specifying turbulence parameters:

Set k and ε explicitly. Set turbulence intensity and turbulence length scale.Set turbulence intensity and turbulent viscosity ratio.Set turbulence intensity and hydraulic diameter.

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Turbulence intensity

The turbulence intensity I is defined as:

Here k is the turbulent kinetic energy and u is the local velocity magnitude.Intensity and length scale depend on conditions upstream:

Exhaust of a turbine. Intensity = 20%. Length scale = 1 - 10 % of blade span.Downstream of perforated plate or screen. Intensity = 10%. Length scale = screen/hole size.Fully-developed flow in a duct or pipe. Intensity = 5%. Length scale = hydraulic diameter.

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Used to bound fluid and solid regions.In viscous flows, no-slip condition enforced at walls.

Tangential fluid velocity equal to wall velocity.Normal velocity component is set to be zero.

Alternatively one can specify the shear stress. Thermal boundary condition.

Several types available.Wall material and thickness can be defined for 1-D or in-plane thin plate heat transfer calculations.

Wall roughness can be defined for turbulent flows.Wall shear stress and heat transfer based on local flow field.

Translational or rotational velocity can be assigned to wall.

Wall boundaries

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symmetry planes

Symmetry boundaries

Used to reduce computational effort in problem.Flow field and geometry must be symmetric:

Zero normal velocity at symmetry plane.Zero normal gradients of all variables at symmetry plane.

No inputs required.Must take care to correctly define symmetry boundary locations.

Also used to model slip walls in viscous flow.

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Periodic boundaries

Used when physical geometry of interest and expected flow pattern and the thermal solution are of a periodically repeating nature.

Reduces computational effort in problem.

Two types available:Δp = 0 across periodic planes.

Rotationally or translationally periodic.Rotationally periodic boundaries require axis of rotation be defined in fluid zone.

Δp is finite across periodic planes.Translationally periodic only.Models fully developed conditions.

Specify either mean Δp per period or net mass flow rate.

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computational domain

Streamlines in a 2D tube heat exchanger

flow direction

Translationally periodic boundaries

4 tangential inlets

Rotationally periodic boundaries

Δp = 0: Δp > 0:

Periodic boundaries: examples

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AXIS boundary

Axis boundaries

Used at the centerline (y=0) of a 2-D axisymmetric grid.

Can also be used where multiple grid lines meet at a point in a 3D O-type grid.

No other inputs are required.

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A fluid zone is the group of cells for which all active equations are solved.Fluid material input required.

Single species, phase.

Optional inputs allow setting of source terms:Mass, momentum, energy, etc.

Define fluid zone as laminar flow region if modeling transitional flow.Can define zone as porous media. Define axis of rotation for rotationally periodic flows.Can define motion for fluid zone.

Cell zones: fluid

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Porous zone modeled as special type of fluid zone.Enable the porous zone option in the fluid boundary conditions panel.Pressure loss in flow determined via user inputs of resistance coefficients to lumped parameter model.

Used to model flow through porous media and other “distributed” resistances, e.g:

Packed beds.Filter papers.Perforated plates.Flow distributors.Tube banks.

Porous media conditions

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For single zone problems use the rotating reference frame model. Define the whole zone as moving reference frame. This has limited applicability.For multiple zone problems each zone can be specified as having a moving reference frame:

Multiple reference frame model. Least accurate, least demanding on CPU.Mixing plane model. Field data are averaged at the outlet of one zone and used as inlet boundary data to adjacent zone.

Or each zone can be defined as moving mesh using the sliding mesh model. Must then also define interface. Mesh positions are calculated as a function of time. Relative motion must be tangential (no normal translation).

Moving zones

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A solid zone is a group of cells for which only heat conduction is solved and no flow equations are solved.The material being treated as solid may actually be fluid, but it is assumed that no convection takes place.The only required input is material type so that appropriate material properties are being used.Optional inputs allow you to set a volumetric heat generation rate (heat source).Need to specify rotation axis if rotationally periodic boundaries adjacent to solid zone.Can define motion for solid zone.

Cell zones: solid

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Internal face boundaries

Defined on cell faces.Do not have finite thickness.Provide means of introducing step change in flow properties.

Used to implement physical models representing:Fans.Radiators.Porous jumps.Interior walls. In that case also called “thin walls.”

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Material properties

For each zone, a material needs to be specified.For the material, relevant properties need to be specified:

Density.Viscosity, may be non-Newtonian.Heat capacity.Molecular weight.Thermal conductivity.Diffusion coefficients.

Which properties need to be specified depends on the model. Not all properties are always required.For mixtures, properties may have to be specified as a function of the mixture composition.

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Fluid density

For constant density, incompressible flow: ρ = constant.

For compressible flow: ρ = pabsolute/RT.

Density can also be defined as a function of temperature (polynomial, piece-wise polynomial, or the Boussinesq model where ρ is considered constant except for the buoyancy term in the momentum equations) or be defined with user specified functions.For incompressible flows where density is a function of temperature one can also use the so-called incompressible-ideal-gas law: ρ = poperating/RT.

Generally speaking, one should set poperating close to the mean pressure in the domain to avoid round-off errors.However, for high Mach number flows using the coupled solver, set poperating to zero.

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When is a problem properly specified?

Proper specification of boundary conditions is very important.Incorrect boundary conditions will lead to incorrect results.Boundary conditions may be overspecified or underspecified.Overspecification occurs when more boundary conditions are specified than appropriate and not all conditions can hold at the same time. Underspecification occurs when the problem is incompletely specified, e.g. there are boundaries for which no condition is specified.Commercially available CFD codes will usually perform a number of checks on the boundary condition set-up to prevent obvious errors from occurring.

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Summary

Zones are used to assign boundary conditions.Wide range of boundary conditions permit flow to enter and exit solution domain.Wall boundary conditions used to bound fluid and solid regions.Repeating boundaries used to reduce computational effort.Internal cell zones used to specify fluid, solid, and porous regions.Internal face boundaries provide way to introduce step change in flow properties.

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Solve

Run CFD code on computer

2D and small 3D simulations can be run on desktop computers (e.g., FlowLab)Unsteady 3D simulations still require large parallel computers

Monitor ResidualsDefined two ways

Change in flow variables between iterationsError in discrete algebraic equation

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Uncertainty Assessment

Process of estimating errors due to numerics and modeling

Numerical errors Iterative non-convergence: monitor residuals

Spatial errors: grid studies and Richardson extrapolation

Temporal errors: time-step studies and Richardson extrapolation

Modeling errors (Turbulence modeling, multi-phase physics, closure of viscous stress tensor for non-Newtonian fluids)

Only way to assess is through comparison with benchmark data which includes EFD uncertainty assessment.

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Conclusions

Capabilities of Current TechnologyComplex real-world problems solved using Scientific ComputingCommercial software available for certain problemsSimulation-based design (i.e. logic-based) is being realized.Ability to study problems that are either expensive, too small, too large, or too dangerous to study in laboratory

Very small: nano- and micro-fluidicsVery large: cosmology (study of the origin, current state, and future of our Universe)Expensive: engineering prototypes (ships, aircraft)Dangerous: explosions, response to weapons of mass destruction

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Conclusions

Limitations of Current Technology For fluid mechanics, many problems not adequately described by Navier-Stokes equations or are beyond current generation computers.

TurbulenceMulti-phase physics: solid-gas (pollution, soot), liquid-gas (bubbles, cavitation); solid-liquid (sediment transport)Combustion and chemical reactionsNon-Newtonian fluids (blood; polymers)

Similar modeling challenges in other branches of engineering and the sciences

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Conclusions

Because of limitations, need for experimental research is greatHowever, focus has changed

From: Research based solely upon experimental observationsBuild and test (although this is still done)

To:High-fidelity measurements in support of validation and building new computational models.

Currently, the best approach to solving engineering problems often uses simulation and experimentation