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

Boundary Conditions

### Type of Boundary Conditions, Applications and Limitations

#### What is physical and mathematical significance of a boundary condition?

The boundary conditions of any problem is used to define the upper and lower limits of the field variables (albeit in absence of any source or sink). These are the operating conditions which govern both the micro- and macro behaviors of these variables. A suitable choice of boundary conditions is as good as a good test set-up! Intuitively, a boundary condition implies that "it is known what happens" on a particular boundary.

There are different (combination) of boundary conditions. For example, in a structural simulation, the number of boundary conditions can be varied to ensure the force- and moment balance of the entire system. This can be achieved by applying boundary condition at just one node or at 6 different nodes! Similarly, in any fluid problem, there must be an entry and an exit for the fluid (as an exception buoyancy-driven flow can be omitted for the time being). This most basic condition is termed as "Inlet" and "Outlet" boundary conditions in CFD parlance, though the choice of "field variables" such as velocity, pressure, temperature, mass flow rate, may vary as per problem set-up.

#### Naming Convection of Boundaries and Cell Zones

In any practical application of CFD simulations, the computational domain may comprise of many cells zones (fluid and solid zones) and boundary zones (walls, inlets and outlets). The engineer responsible for pre-processing may not be the one who creates solver file and post-processes the results. The reviewer(s) of the mesh and simulation set-up will certainly be not the engineer who created them. In order to convey the domain information seamlessly, a naming convention should be adopted, it can be a generic system applicable for large number of projects or a specific system for particular simulation set-up. An example is outlined below with following default setting: Newtonian, stationary, adiabatic, smooth boundaries or zones can be named arbitrarily though it is recommended to chose names and identifiers meaningfully.
1. Inlet(s) and outlet(s) should be named as b_inlet_id, b_outlet_id where 'id' stands for identifier.
2. Walls with boundary conditions (other than adiabatic): w_tpr_id, w_hfx_id, w_htc_id, w_rot_id w_mov_id, w_rgh_id, w_s2s_id where 'rgh' stands for rough walls, s2s stands for coupled solid walls.
3. Cell zones: fld_air_id, fld_wtr_id, fld_oil_id, fld_nnw_id, fld_rot_id, por_mom_id, por_thm_id, sld_alm_id, sld_stl_id, sld_src_id ... where 'nnw' stands for non-Newtonian fluids, 'src' stands for solid zones with heat source and/or sink and 'por' stands for porous (fluid) zones.
4. Internal planes: int_f2f_id, int_baf_id ... where 'baf' stands for thin walls or baffles.
5. Non-conformal interfaces: if_f2f_id, if_s2s_id, if_sta_rot_id, if_por_fld_id, if_por_por_id where 'sta' stands for 'stationary' zones and 'por' stands for 'porous' zones.
6. All boundaries with no special setting or conditions to be applied on them can be named as 'b_def_id'.

#### Inlet

This is the 1st member of the pair of boundary conditions which are must for any CFD calculations in a forced convection situation. Of course a natural convection case does not required any inlet or outlet. The primary consideration of an inlet B.C. is to select between the Mass Flow Rate, Static Pressure and Total Pressure based on the actual information available about the operating conditions of the system and robustness of the solver, (the matrix inversion) algorithm which keeps running till solution is achieved. While tempting to use velocity inlet B.C. care needs to be taken to account for change in cross-sectional area when an arc is represented by a set of connected lines.

The sign convention for mass flow is such that a positive value represents flow into the domain and a negative value represents flow out of the domain. Sometimes an 'outlet' can be specified as inlet with negative velocity to represent some specified mass flow rate.

Some other considerations during application of Inlet B.C. is "Fully Developed Flow" Vs "Developing Flow". For example, if you are a beginner learning tips and trick of CFD by trying to simulate HTC and correlating it with Dittus-Boelter equation, make sure that the flow regime is fully developed. Sometimes, the inlet of the problem set-up is moved upstream the actual location to get the flow a bit developed. Specification of turbulence parameters (turbulent kinetic energy, TKE and turbulent eddy dissipation, TED) should be based on actual measurement of as far as possible. When there are any source of momentum such as centrifugal fan in the computation domain or sharp edges, the overall result gets affected by the turbulence set at the inlet. Followings are the methods to specify turbulence:

• Specify TKE [m2/s2] and TED [m2/s3] explicitly
• Turbulent Intensity [%] and Turbulent Viscosity Ratio (TVR) [-]
• Hydraulic Diameter [m] and Turbulent Intensity [%]
• Turbulent Intensity [%] and Length Scale [m]
These requirements on turbulent parameters further depends on flow type: external or internal. The length scale 'L' For external flows is typically the length scales along the flow direction.
• External Flows
• Length Scale = 0.07 x L
• Turbulent Intensity: Based on upstream condition
• Turbulent Viscosity Ratio: 1 < TVR < 10
• Internal Flows
• ~ Length Scale = Hydraulic Diam.
• Turbulent Intensity: 0.16 x Re-1/8
• Turbulent Viscosity Ratio: 1 < TVR < 10
The setting for inlet and outlet boundary conditions in ANSYS FLUENT is shown below:

#### Outlet

Boundary source: Inlet can also have a boundary source define to model heat sources such as solar radiation.

#### Wall Boundary - What is physical and mathematical significance?

Walls are required to store a liquid or contain the expansion (mixing) of gases. Since all the fluid flow has to be contained inside walls or at least in a channel, wall B.C. is natural extension into the numerical simulation process. Wall are not only the source of 'turbulence' that gets generated in the flow domain, its surface characteristics becomes important if certain assumptions gets violated. In any CFD software it is not necessary to create 'named' 2D regions for the walls. This is because any faces of a 3D region which do not explicitly have a 2D region assigned to them, are automatically assigned to the default B.C. 'wall' having 'Adiabatic' condition. In case one wishes to create walls such as "Isothermal / Rotating / Heat Flux Wall", it must be created during the pre-processing. Typically, there is no flow across the wall boundary conditions. However, in case of permeable or porous walls, flow does occur across the wall. Similarly, in case of suction or blowing (for example transpiration cooling in Gas Turbine Blades), the mass flow rate specifications are required on the wall boundary conditions.

The setting for wall boundary conditions in ANSYS FLUENT is shown below:

Typical classification of wall B.C. is:
• No-slip: Velocity of fluid at wall boundary is same as fluid velocity.
• Free slip: Velocity component parallel to wall has finite value (computed by the solver), but the velocity normal to the wall and shear stress both set to zero. Zero gradients for other field variables are not enforced in slip wall conditions.
• Wall Roughness: Walls are assumed to be hydraulically smooth so long the "sand roughness height" is inside the Laminar Sub-layer. Roughness is also called "Rugocity". Typically roughness is caused by small protrusions over the mean surface of a manufactured component. Any such "technical roughness" can be converted into a "equivalent sand roughness".
• ks = Sand Roughness Height [m] or [μm], k+ = ks/h where h is characteristic height of viscous sub-layer. Please refer to the turbulence modeling page for definition of viscous sub-layer.
• ks >> h: In this case roughness element take up all of the boundary layer and hence the viscosity is of no further importance (also call "Fully Rough Regime" where flow is independent of Reynolds Number.
• ks < h: Here roughness elements are still completely within the purely viscous sub-layer and the flow can be assumed to be "hydraulically smooth", that is, there is no difference as compared to the ideal smooth surface.
• Roughness Reynolds number, Reks is defined as Reks = ρ.u+.ks/μ and the surface roughness condition can be defined as:
• Reks ≤ 5: hydraulically smooth and the wall surface roughness need not be activated in CFD simulations
• 5 < Reks ≤ 70: transitionally smooth and wall surface roughness can be activated in CFD simulations though the impact of pressure drop or wall shear stress may not always be noticeable
• Reks > 70: fully rough regime and wall surface roughness need to be activated in CFD simulations.
• Contact Resistance: By default, the walls are assumed to have zero thickness. This setting can be used specify the typical contact resistance between fluid-solid such as fouling factors or the contact resistance between solid-solid. If the thickness of the wall needs to be modeled to account for conductive resistance normal to the plane only (no conduction along the plane), contact resistance value can be specified as [t/k] where t = thickness of the wall and k is thermal conductivity of the solid.

Some typical value of thermal conductance when air gaps are not evaluated are tabulated below (reference: A Textbook fo Heat Transfer by Lienhard IV and Lienhard V) -  Surface Pair Thermal Conductance [W/m2K] Copper - Copper 10,000 ~ 25,000 Aluminium - Aluminium 2,200 ~ 12,000 Ceramic - Metals 1,500 ~ 8,500 Stainless steel - Stainless steel 2,000 ~ 3,7000 Ceramic - Ceramic 500 ~ 3,000
Thus, for a ceramic - aluminium pair (such as in electronic chips) with contact surface area of 20 [mm] x 20 [mm], the thermal contact resistance would be in the range 1.67 ~ 0.30 [K/W]. Note that contact resistance is also a function of contact area, higher the surface area lesser ther contact resistance in [K/W]. For a ceramic - ceramic pair having contact surface area of 10 [mm] x 10 [mm], the thermal contact resistance would be in the range 20.0 ~ 3.33 [K/W].
• Shell Conduction: If a thickness is specified to a wall then thermal resistance across the wall thickness is imposed though conduction is considered in the wall in the normal direction only. If the walls are of nearly uniform thickness and the conductive heat transfer needs to be accounted for both in-plane and through-the-plane conduction, this feature can be used. The user needs to specify only the thickness and thermal conductivity of the solid as described in case of contact resistance.

Shell conduction can be used to account for thermal mass in transient thermal analysis problems such as thermal soaking (ramp-up) or thermal cool-down. It can also be used for multiple junctions and allows heat conduction through the junctions. Shell conduction can be applied on boundary walls as well as internal walls. Fluxes at the ends of a shell conducting wall are not included in heat balance reports. These fluxes are accounted for correctly in the ANSYS FLUENT solution, but are not listed in the flux report.

FLUENT creates a shadow wall by adding suffix ':external' to the walls on which shell conduction is switched ON.

• Wall Motion:

• Radiation: Walls can be defined as opaque, semi-transparent or transparent to model radiation effects. An opaque wall participate by absorbing, reflecting and emitting radiation. A semi-transparent wall will transmit radiation through it as well. On the other hand, these properties of a solid may be different for infra-red radiation and solar radiation. For example, glass is transparent to solar radiation (that is solar radiation can pass through a glass wall without any reflection/absorption) whereas it is opaque/semi-transparent to infra-red radiation (it traps these waves by refection and absorption).
• DPM (Discrete Particle Modeling): In DPM simulations dealings with solid-liquid or solid-gas where particles or droplets interact with walls, the boundary conditions need to be specified as per expected behaviour. When a solid particle hits the wall, it may either reflect or get captures (stick to the wall). Elastic impact is not realistic even at particles with diameters at micron level. Hence, an appropriate coefficient of restitution need to be specified. Usually, particles traveling at lower velocity (how much lower - it is not a unique value) and smallers ones tend to get captured by the wall. The wall-particle interaction phenomena results in deposition of dust particles on solar panels, leaves of the trees, window panes and the blades of a rotating ceiling fan. Gravity can also have significant effect on dust deposition for particles with diameters > 50 [μm]. For liquid droplets, there are 4 different types of wall-droplet interactions: Splash, Stick, Rebound and Breakup. The rebound with known coefficient of restitution in normal direction (EN) and tangential direction (Eθ) is explained in following diagram.

#### Periodic Boundary Conditions

Strictly speaking, this is not a boundary condition. That is, any numerical simulation can proceed without it. However, this is a great tool to reduce the computational effort and resource if the flow can be envisaged to be symmetrical about a plane or pair of planes. It must be noted that there is a subtle difference between geometrical symmetry and periodicity. Periodic interfaces are treated as if one side of the interface has been translated or rotated to align with the second side of the interface. The periodic type determines the type of transformation (translational or rotational) used to map one side of the interface to the other.

• They must be in pairs.
• They have to be physically identical.
• There is a symmetry. But, unlike a symmetry BC, there is a flow normal to the BC
• The flow field in at one BC is equal to the flow field out at the other
• Types of periodic boundaries
• Transnational Periodic BC: In this case the two sides of the interface must be parallel to each other such that a single translation transformation can be used to map Region List 1 to Region List 2. Flow around a single louver in a whole array in a heat exchanger fin is an example
• Rotational Periodic BC: In this case the two sides of the periodic interface can be mapped by a single rotational transformation about an axis. Flow domain through an Axial Flow Fan can be reduced using rotational periodic B.C.

#### Symmetry Boundary Conditions

Strictly speaking, this also is not a boundary condition. That is, any numerical simulation can proceed without it. However, this is a great tool to reduce the computational effort and resource if the flow can be envisaged to be symmetrical about a plane or pair of planes. It must be noted that the geometrical symmetry does not guarantee symmetry of the flow. Similarly, cases where micro-structure of flow eddies are being captured such "Large Eddy Simulation" or "DES – Detached Eddy Simulation", symmetry cannot be used owing to inherent 3D nature of the eddies.

• By definition, a symmetry BC refers to planar boundary surface. If 2 surfaces which meet at a sharp angle & both are symmetric planes, set each surface to be a separate named boundary condition, rather than combine them into a single one.
• Velocity component normal to the Symmetry Plane Boundary = 0. Scalar variable gradients normal to the plane is also =0
• If a particle reaches symmetry plane, it is reflected back.
• Symmetric geometry doesn't necessarily imply that the flow field is also symmetric. For example, a jet entering at the centre of a symmetrical duct will tend to flow along one side above a certain Reynolds number. This is known as the Coanda effect. If a symmetry plane is this situation, an incorrect flow field will be obtained.

##### Heat Exchanger Modeling

Except shell and tube heat exchangers, for most engineering problems it is impractical and sometimes impossible to correctly model individual fins and tubes of a heat exchanger core. In cross-flow heat exchangers such as automotive radiators, air is called primary fluid (the heat sink) and water or coolant as auxiliary (secondary) fluid (the heat source). There are multiple approaches to model such heat exchangers when combined with surrounding systems and sub-systems.

The heat exchanger models were designed for 'compact' heat exchangers, implying that the primary fluid side flow is unidirectional. The auxiliary fluid is assumed to flow through a large number of parallel tubes, which can optionally double back in a serpentine pattern to create a number of passes. One can independently choose the direction of principal auxiliary fluid flow, the pass-to-pass direction and the direction of external primary fluid flow. Pressure loss is modeled as a momentum sink in the momentum equation and heat transfer is modeled as a heat source in the energy equation.

 Lumped Model Macro Model Dual Cell Model Accounts for the pressure loss and auxiliary fluid heat rejection Simple-effectiveness-model Number-of-Transfer-Units (NTU) model Number-of-Transfer-Units (NTU) model Core modeled as volumetric heat source Compute auxiliary fluid inlet temperature for a fixed heat rejection or total heat rejection for a fixed auxiliary fluid inlet temperature NTU method for heat transfer calculations - Auxiliary flow is modeled as 1D flow Auxiliary flow modelled on a separate mesh (other than the primary fluid mesh)

##### CFX Recommendation on pair - combination of boundary conditions
 Solver Behaviour Inlet Outlet Most Robust Velocity or Mass Flow Rate Static Pressure Somewhat Robust Total Pressure Velocity or Mass Flow Rate Sensitive of Guess (Initialization) Total Pressure Static Pressure Unreliable Static Pressure Static Pressure Not possible (divergence guaranteed) Any Total Pressure
##### FLUENT Recommendation on pair - combination of boundary conditions
 Solver Behaviour Inlet Outlet Most Robust Velocity or Mass Flow Rate Static Pressure Somewhat Robust Velocity or Mass Flow Rate Outflow or Outlet-vent Only for incompressible flows Velocity Inlet Outflow Not available Any Mass Flow Rate Not for compressible Specified Velocity Any

#### FAN Boundary Conditions

A fan is considered to be infinitely thin, and the discontinuous pressure rise across it is specified as a function of the velocity through the fan. The relationship may be a constant, a polynomial - of the form a + b*x2 + ... , or piecewise-linear, or piecewise-polynomial function, or a user-defined function. In FLUENT, a zero thickness plane can be used to represent a fan. However, in CFX a volume is required to represent fan as momentum source coefficient. The general momentum source in CFX has unit of [kg/m3-s2]. The source can be linearlized for better convergence and stability which has unit of [kg/m3-s].

• Fan should be modelled so that a pressure rise occurs for forward flow through the fan.
• Since the fan is considered to be infinitely thin, it must be modeled as the interface between cells, rather than a cell zone. Thus the fan zone is a type of internal face zone (where the faces are line segments in 2D or triangles/quadrilaterals in 3D).
• Thun when mesh is read into ANSYS FLUENT, the fan zone is identified as an interior zone.
• You can use the Surface Integrals dialog box to report the pressure rise through the fan as described by following steps.
• Create a surface on each side of the fan zone - just upstream and downstream to create two new surfaces.
• In the Surface Integrals dialog box, report the average Static Pressure just upstream and just downstream of the fan. The pressure rise through the fan is difference of downstream and upstream values.
• While generating contour plots, turn off the display of node values to see the different values on each side of the fan. If node values are displayed, the cell values on either side of the fan will be averaged to obtain a node value, and you will not see distinct (pressure, temperature, velocity ...) values on the two sides of the fan.

#### POROUS JUMP Boundary Conditions

This is opposite to the fan boundary conditions define above and like fan is also is considered to be infinitely thin membrane, and the discontinuous pressure drop across it is specified as a function of the velocity through the fan. The relationship may be a constant, a polynomial - of the form a + b*x2 + ... , or piecewise-linear, or piecewise-polynomial function, or a user-defined function. There is no similar boundary conditions available in CFX.

• It should be modelled so that a pressure drop occurs for forward flow through the porous jump.
• Porous jump should be used (instead of the full porous media model) whenever possible because it is more robust and yields better convergence.
• The porous jump model is applied to a face zone, not to a cell zone.
• By default ANSYS FLUENT uses and reports a superficial velocity inside the porous medium, based on the volumetric flow rate, to ensure continuity of the velocity vectors across the porous medium interface.
• You can use the Surface Integrals dialog box to report the pressure drop through the POROUS JUMP membrane as described by following steps.
• Create a surface on each side of the fan zone - just upstream and downstream to create two new surfaces.
• In the Surface Integrals dialog box, report the average Static Pressure just upstream and just downstream of the porous jump membrane. The pressure drop through the membrane is difference of upstream and downstream values.
• While generating contour plots, turn off the display of node values to see the different values on each side of the fan. If node values are displayed, the cell values on either side of the fan will be averaged to obtain a node value, and you will not see distinct (pressure, temperature, velocity ...) values on the two sides of the fan.

#### Porous Domains

Flow geometry such as heat exchangers with closely spaced fins, honeycomb flow passages in a catalytic converters, screens or perforated plates used as protection cover at the from of a tractor engine ... are too complicated to model as it is. They are simplified with equivalent performance characteristic, knowns as Δp-Q curve. These curves are either generated using empirical correlations from textbooks or using a CFD simulation for smalled, periodic / symmetric flow arrangement. The simplified computational domain is known as "porous zone" in case it is represented as a 3D volume or pressure or porous jump in case it is represented as a plane of zero thickness. In a similar fashion, the performance data of a fan can be specified including the swirl component.

Examples of porous materials: note that a material which does not allow convection of fluid (e.g. brick, soil, cotton...) may allow diffusion of the fluid (brick, cotton...) such as water. In general, most of the applications of porous domain in CFD simulations refer to convection phenomena such as flow through the radiator core and honeycomb in a catalytic converter.

All the porous media formulation take the form: Δp = -L × (A.v + B.v2) 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 and CFX, the equation used is: Δp/L = -(μ/α.v + C2.ρ/2.v2) where α is known as 'permeability' and μ is the dynamic viscosity of fluid flowing through the porous domain. Note that FLUENT takes 1/α as input for porous regions. This is a measure of flow resistance and the unit of permeability (α) is [m2]. Other unit of measurement is the darcy [1 darcy = 0.987 μm2], named after the French scientist who discovered the phenomenon. The unit of inertial loss coefficient (C2) is [m-1]. Inertial loss coefficient is known as Quadratic Loss Coefficient in CFX. An alternate formulation in CFX is in terms of Linear Loss Coefficient [kg/m3-s] and Quadratic Loss Coefficients [kg/m4] which are defined as A and B respectively above or Pv and Pi below.

STAR-CCM+ uses the expression Δp/L = -(Pv.v + Pi.v2) for a porous domain.

The pressure drop is usually specified as Δp = ζ/2·ρ·v2 where ζ is 'equivalent loss coefficient' and is dimensionless. Darcy expressed the pressure gradient in the porous media as v = -[K/μ]·dP/dL where 'K' is the permeability and 'v' is the superficial velocity or the apparent velocity determined by dividing the flow rate by the cross-sectional area across which fluid is flowing. All the programs FLUENT, CFX and STAR-CCM+ uses superficial velocity.

Steps to find out viscous and inertial resistances:

1. Calculate the pressure drop vs. flow velocity data [Δp-v] from empirical correlations or wind-tunnel test or simplified CFD simulations.
2. Divide the pressure drop value with thickness of the porous domain. Let's name it as [Δp' vs. v curve].
3. Calculate the quadratic polynomial curve fit coefficients [A, B] from the curve Δp' vs. -v. Ensure that the intercept to the y-axis is zero.
4. In STAR-CCM+ these coefficients 'A' and 'B' can be directly used as Pv and Pi which are viscous and inertial resistance coefficients respectively.
5. 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".
6. Divide 'B' by [0.5ρ] where ρ is the density of fluid, to get C2 to be used in ANSYS FLUENT as "inertial resistance coefficient".
7. 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.
The GUI to set porous domain in ANSYS FLUENT is as shown below. For 3D cases, direction vectors for any two principal axes need to be specified, the third direction is automatically calculated by FLUENT. However, one must be consistent in specification of direction vector and resistance coefficients.

When pressure drop is linear function of velocity, it is known as Darcy's Law: Δp = - μ/α × v. This option is implemented explicitly in COMSOL.

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:

Porous Loss Coefficient Calculator

Note that the loss coefficients are assumed to be (and most of the time they are so) parabalic. Hence, only 3 sets of points are required to derive the parabolic equation. It is recommended that the first point is close to no-flow conditions, second point close to the mid-point of operation and the last point towards the highest flow rate conditions. In the table below, Pi refers to pressure drop across the porous domain at Vi. Air is assumed to be the working fluid. Sutherland law is used to find out viscosity of air at specified temperature.

 Thickness of HX TH [mm] First point V1 [m/s] First point P1 [Pa] Second point V2 [m/s] Second point P2 [Pa] Third point V3 [m/s] Third point P3 [Pa] Gas Temperature Tg [°C] Gas Pressure Pg [kPa]

Example calculation: x = {0.0   1.0   2.5}, Y = {0   50   75}, m1 = 50.0, m2 = 16.67, A = -13.33, B = 63.33, C = 0.0. The parabola generated by the curve fit coefficients looks like as shown below. Note that a parabola with local maxima towards higher value of X-axis may have a negative value of A or B.

The formula to derive A, B and C are as follows:

###### Examples of Porous Media
One of the frequent application of porous media formulation is the heat exchangers such as automotive radiators, inter-coolers, oil-cooler and honeycombs in catalytic converters. The porous formulations can also be used in Fabric Ducts used in HVAC systems. These ducts an either be "permeable only textile fabric where air is squeezed between loose fibres" or ducts with micro-perforations with laser cut holes as low as 0.2 [mm]. Micro-perforations allow creation of variety of low velocity air patterns to suit local cooling needs. In porous fabric ducts, air is allowed to pass through the fabric with the air-flow rate controlled by the fabric weave and the internal static pressure. In non-porous fabric, no air can pass through the fabric weave and air-flow is delivered through orifices and other venting options (nozzles, micro-perforations) to achieve desired air-flow.

Porosity provides information on the overall pore volume of a porous material and is defined as the ratio of the nonsolid volume (voids) to the total volume of the nonwoven fabric. The air permeability and porosity of fabric duct are inter-dependent. An increase in porosity (the free space) should lead to decrease in permeability (resistance to the air flow). From flow point of view, a volume based definition of porosity may not provide a direct relation to the the flow velocity which depends on flow area perpendicular to the flow direction.

###### Atmospheric Data
Reference: Analytic Combustion by Anil W. Date (Cambridge Press).

There is decrease in atmospheric pressure and temperature with altitude as compared to height above sea level. Why sea level is considered as reference datum? This is because the lquids maintain uniform level and any point anywhere in the sea is expected to be same radial distance from the centre of the Earth.

p [Pa] = 101325 * (1 - 2.25577E-05 × H)5.2559 where altitude H is in [m].

T [K] = 288.15 - 0.0065 × H. You may use the following calculator to estimate ambient pressure and tempearture at higher altitudes. There is option to chose altitude in [m] or [ft]. However, the outputs are in SI units.

 Altitude, H Unit [m] [ft]

###### Binary Diffusion Coefficients

In situations with multi-component flows (such as leakage of fuel or refrigerant) where diffusion dominates the correct specification of binary diffusion coefficient is very important. Following table specifies value at 1 [atm] and 300 [K]. Reference: Analytic Combustion by Anil W. Date.

 Pair Dab [m2/s] H2O - air 24.0E-6 CO - air 19.0E-6 CO2 - air 14.0E-6 H2 - air 78.0E-6 O2 - air 19.0E-6 SO2 - air 13.0E-6 NH3 - air 28.0E-6 CH3OH - air 14.0E-6 C2H5OH - air 11.0E-6 CH4 - air 16.0E-6 C6H6 - air 8.00E-6 C8H18 - air 5.00E-6 C8H16 - air 7.10E-6 C10H22 - air 6.00E-6 O2 - H2 70.0E-6 CO2 - N2 11.0E-6 CO2 -H2 55.0E-6 C6H14 - N2 8.00E-6 C8H18 - N2 7.00E-6 C10H22 - N2 6.40E-6

###### Darcy Law for Porous Media
This is the basic law governing the flow of fluids through porous media such as soil, rocks and sand beds. This is analogous to other linear phenomenological transport laws namely Ohm’s law for electrical conduction, Fick’s law for solute diffusion and Fourier’s law for heat conduction. Note that Darcy’s law is a macroscopic law will hold true over regions that are much larger than the size of a single pore.

Here:
• Q = Volumetric flow rate [m3/s]
• A = cross section area of the flow passage [m2
• L = Length of flow path along the direction of flow [m]
• ΔP = pressure drop along the direction of flow = [p - ρgh] [Pa], ρ = density of fluid [kg/m3], g = acceleration due to gravity [m2/s], h = height along the direction of gravity [m]
• C = constant of proportionality [m2/Pa.s] = μ/k, μ = dynamic viscosity of fluid [Pa.s], k = permeability [m2]
In petroleum engineering, due to very low permeability of rocks, 'Darcy' unit defined by 1 [Darcy] = 0.987×10-12 [m2] is widely used. The Darcy unit can be interpreted as a flow rate of 1 [ml/s] through a rock of fluid with viscosity 1 [cP] = 0.001 [Pa.s] through a cross-section of of 1 [cm2] when the pressure drop along the direction of flow were 1 [atm/cm].

Dupuit-Thiem equation (based on Darcy Law in cylindrical coordinate system) is a widely used formula to estimate pressure drop across the wall for a known (oil extraction) flow rate in a circular reservoir that has a constant pressure at its outer boundary.

Dust Accumulation in Air Filters: There are many application of air filters such as automotive air cleaners. Dust Holding Capacity (DHC) is one of the key parameters of such filters. The filter are orthotropic porous media where the porous loss coefficients are different along the 3 directions. However, any CFD simulations to deal with dust accumulation will be a transient simulation where the behaviour of porous domain will change depending upon duct collection level and spatial distribution. This is because filter may not collect dust uniformly and hence permeability will change non-uniformly. For most practical applications, change in pressure drop can be assumed to be a linear function of duct loading (the amount of dust trapped in filters). How does one model the trapping of dust particles in the pores of the filter? Neither the filter pores nor the diameters of the particles are uniform in size and shape!

###### Convergence Troubleshooting Strategies for Porous Media

The rate of convergence slows a porous region is defined such that pressure drop is relatively large in the flow direction (e.g. the permeability is low or the inertial factor is large). This slow convergence can occur because the porous media pressure drop appears as a momentum source term yielding a loss of diagonal dominance in the matrix of equations solved. The best remedy for poor convergence of a problem involving a porous medium is to supply a good initial guess for the pressure drop across the medium. You can supply this guess by patching a value for the pressure in the fluid cells upstream and/or downstream of the medium. It is important to recall, when patching the pressure, that the pressures you input should be defined as the gauge pressures used by the solver (i.e. relative to the operating pressure defined in the simulation).

Another possible way to deal with poor convergence is to temporarily disable the porous media model and obtain an initial flow field without the effect of the porous region. Once an initial solution is obtained, or the calculation is proceeding steadily to convergence, enable the porous media model and continue the calculation with the porous region included. (This method is not recommended for porous media with high resistance.)

Simulations involving highly anisotropic porous media may, at times, pose convergence troubles. This issue can be addressed limiting the anisotropy of the porous media coefficients to two (102) or three (103) orders of magnitude. Even if the medium's resistance in one direction is infinite, it is not needed to set the resistance in that direction to be greater than 1000 times the resistance in the primary flow direction.

##### Tortuosity
Tortuosity - derived from work 'tortuous' - is a measure of the geometric and flow path complexity of a porous medium. A molecule often has to traverse a path that is several times longer than the straight line between its original source and destination. Tortuosity is a ratio that characterizes the tortuous and meandering (convoluted) pathways of fluid convection and/or diffusion through the media.

In the fluid mechanics of porous media, tortuosity is the ratio of the length of a streamline to that of the straight-line distance between those points. A measure of deviation from a straight line. It is the ratio of the actual distance traveled between two points, including any curves encountered, divided by the straight line distance. Tortuosity is used by drillers to describe wellbore trajectory, by log analysts to describe electrical current flow through rock and by geologists to describe pore systems in rock and the meander of rivers.

A related concept is fractal which is used to describe the effective length of rivers and used even for trading in stock markets.

##### Liquid Jet Break-up
[1]:The near- and far-field break-up and atomization of a liquid jet (aerodynamic break-up of liquids) in multi-phase fluid systems fits in the broader category of multi-phase fluid flow. Secondary liquid break-up and coalescence of the droplets in the far-field atomization of the jet is also important in many application such as Diesel sprays, two-phase flow combustion and propulsion problems. The destabilization of the liquid jet close to the nozzle exit is a Kelvin–Helmholtz type of instability where surface tension acts as a stabilizing force and imposes a lower cut-off for the waves which can grow. This cut-off is given by a critical Weber number. Three forces are involved in the splitting of the liquid, namely surface tension, viscous and inertia forces.

[2]: A method for aerodynamic drop break-up that has been widely implemented is known as the Taylor Analogy Break-up model or TAB. The TAB model approximates a drop as a damped oscillator system and tracks a deformation parameter. When the deformation parameter is sufficiently large, break-up is assumed.

##### Surface Tension and Capillary Effect
Though CFD may not be required to solve phenomena such as capillary rise, any flow geometry where surface tension effects are comparable to viscous effects should be dealt carefully. Following chart and OCTAVE (or MATLAB) scrip summarizes the effect of pipe diameter on capillary rise and volume of liquid that can be lifted.

Some historical notes: Geovanni Borelli (1608-1675) demonstrated experimentally that h ∝ 1/r. James Jurin (1684-1750), an English physiologist who independently confirmed that h ~ 1/r and hence the capillary rise law is also known as Jurin’s Law. As the water rises in tube, the total energy of system is sum of "surface energy" and "gravitational potential energy".
• σ: surface tension of liquid which is measure of cohesive force between liquid molecules.
• H: Capillary rise (or depression) - lower point of the meniscus. Note that the capillary effect is the net effect of competitive forces adhesion (force between liquid and solid molecules) and cohesion. Contact angle is a constant property of liquid-solid interface and affects capillary rise.
• If contact angle is zero, the liquid surface is parallel to solid surface and the liquid is said to wet the solid completely. The equation relating to the contact angle and surface tension between all 3 interfaces namely liquid-solid, liquid-gas and solid-gas is known as Young's equation. σSL - σSG + σLG cos(θ) = 0 where S, L and G refer to solid, liquid and gas phases.
• System energy, E = σ × (2πrH) + ρ/2×(πr2H) × g
• Capillary length, LC = [2σ/ρ/g]0.5 ~ 4.0 [mm] for water at room temperature.
• Under dynamic condition when liquid level is increasing in the capillary tube, its rise is resisted by a combination of gravity, viscosity, fluid inertia and dynamic pressure.
• The timescale required to establish Poiseuille flow is, t = 4ρr2/μ where μ is the dynamic viscosity. For water and 0.50 [mm] tube: t ~ 1.2 [s], for 0.20 [mm] tube: t = 0.2 [s]. If rise timescale is less than this value, inertia of liquid mass dominates and inertial overshoot results in oscillation of liquid column about steady state (equilibrium) height.
• Note that the capillary rise predicted is 15 [m] for micro-pores (r =1.0 μm ie. 1.0E-6 m).
• The nature of wetting depends on the choice of liquid as well as on the nature of the surface. For example, water spreads on a clean glass surface but beads up on a glass sheet coated with a monolayer of dimethyloctylchlorosilane (generating a hydrophobic surface).
• Spreading of liquid also depends on the nature of the surrounding fluid. For example, oil droplets on a surface under water have a different contact angle than an oil drop in air. In both the cases, the fluid pairs are immisible.

Surface tension and angle of contact with water and some of the non-metals are tabulated below as per www.accudynetest.com.

 Surface Tension of Water with Abbreviation [N/m] [°] Silicone Oxide Glass 0.0725 ~ 0 Poly-Vinyl-Chloride PVC 0.0379 85.6 Poly-Tetra-Fluoro-Ethylene PTFE 0.0194 109 Poly-Amide-6-6 Nylon-66 0.0422 68.3 Poly-Methyle-Meth-Acrylate PMMA (Acrylic, Plexiglass) 0.0375 70.9 Poly-Ethylene-Terephthalate PET 0.0390 72.5 Poly-Carbonate PC 0.0440 82.0 Acrylonitril Butadine Styrene ABS 0.0385 80.9 Poly-Ethylene PE 0.0316 96.0 Poly-Propylene PP 0.0305 102

###### OCTAVE Script
Note that the script does not check whether the radius of the capillary is << capillary length or not. The increase in volume of liquid with increasing radii of the capillary tube is counter-intutive. Can you explain why this behaviour is observed?
```%Script to calculate capillary rise and plot a curve for different radii

%Surface Tension [N/m]
%Water (at 20 C): 0.073, Glycerin: 0.063, Blood (at 37 C): 0.058, Ammonia: 0.021
%Ethyl alcohol: 0.023,	 Kerosine: 0.028, Soap solution: 0.025,   Mercury: 0.440
s = 0.073;

%Contact angle (deg) - depends on liquid-solid combination
q = 0;   %For water-glass combination

%Density [kg/m^3]
rho = 990;

%Acceleration due to gravity [m/s^2]
g = 9.806;

R1 = 0.25;

R2 = 1.25;

%------------------------------------------------------------------------------
dr = (R2 - R1)/25;
r = [R1: dr: R2];
h = (2000 * s * cos(q*pi/180) / rho / g ./ r) * 1000;
V = pi .* r .^2 .* h / 1000;

hold on; subplot(311);
plot(2*r, h, "linestyle", ":", "linewidth", 2, "marker", "o");
xlabel('Tube Diameter, d [mm]');
ylabel('Capillary Rise [mm]'); grid on;
%
% Format X-axis ticks
xtick = get (gca, "xtick");
xticklabel = strsplit (sprintf ("%.1f\n", xtick), "\n", true);
set (gca, "xticklabel", xticklabel)
%
% Format Y-Axis ticks
ytick = get (gca, "ytick");
yticklabel = strsplit (sprintf ("%.1f\n", ytick), "\n", true);
set (gca, "yticklabel", yticklabel);
%
subplot(312);
plot(2*r, V, "linestyle", ":", "linewidth", 2, "marker", "o");
xlabel('Tube Diameter, d [mm]');
ylabel('Capillary Volume [mL or cm^3]'); grid on;
%
% Format X-axis ticks
xtick = get (gca, "xtick");
xticklabel = strsplit (sprintf ("%.2f\n", xtick), "\n", true);
set (gca, "xticklabel", xticklabel)
%
% Format Y-Axis ticks
ytick = get (gca, "ytick");
yticklabel = strsplit (sprintf ("%.2f\n", ytick), "\n", true);
set (gca, "yticklabel", yticklabel);
%
subplot(313);
plot(2*r, V*1000, "linestyle", ":", "linewidth", 2, "marker", "o");
xlabel('Tube Diameter, d [mm]');
ylabel('Capillary Volume [\muL]'); grid on;
%
% Format X-axis ticks
xtick = get (gca, "xtick");
xticklabel = strsplit (sprintf ("%.2f\n", xtick), "\n", true);
set (gca, "xticklabel", xticklabel)
%
% Format Y-Axis ticks
ytick = get (gca, "ytick");
yticklabel = strsplit (sprintf ("%.0f\n", ytick), "\n", true);
set (gca, "yticklabel", yticklabel);```
Have you noticed why the water does not spill even when the water level is above the brink of a bowl!

One of the applications of capillary effect combined with fully-developed laminar (Poiseuille) flow is pipetting. Pipetting process is aspiration of a pre-determined volume of liquid by creating a vacuum above a tapered capillary tube (known as tip). The pressure in the pipette chamber during the process is in a dynamic equilibrium and is affected by the ambient pressure, viscosity, surface tension and density of the liquid, and the speed of the piston movement. The suction (aspiration) of liquid in pipette tips normally undergo following 4 phases:

• Acceleration phase: The rate of decrease of pressure the pipette chamber is higher than the rate of increase of pressure caused by reduction in gas volume due to aspirated volume of the liquid. Thus, the fluid-gas interface will tend to accelerate.
• Uniform speed phase: The pressure in the cavity reduced uniformly and is balanced by actual pressure increase due to reduction in gas volume caused by suction of fluid.
• Deceleration phase: the piston speed slows down, but the total pressure difference between the inside and outside of the pipette chamber is still high to keep the fluid moving. The liquid suction speed gradually slows down and finally maintains the balance.
• Balancing phase: the pipetting operation is completed and the static equilibrium is achieved between pressure, surface tension and hydrostatic forces.

 Regime-I Regime-II Regime-III Regime-IV Initial boundary effects important Viscous effect negligible Poiseuille flow: inertia effect negligible Late viscous regime Surface tension forces dominant Capillary rise resisted by fluid inertia Lucas-Washburn law applies Fluid rise approaches steady state height Capillary rise: z ∝ t2 Capillary rise: z ∝ t Capillary rise: z ∝ t0.5 Capillary rise: z ∝ e-t

##### Mesh and Simulation Set-up File Review
One of the challenges for any reviewer is to understand the geometry and the naming convention used for define boundary and fluid zone. Just looking at the name of the zones, no information can be gathers if it is arbitrarily designated. Some naming convention will go a long way in making the review process easier. One of the many ways is:
1. Use of prefixes and suffixes appropriately
2. Name fluid zones as f-air- or f-water- for fluids, s-steel- for solids, por- for porous.
3. Add bc-inlet-tpr for "Total Pressure" boundary condition at inlet, bc-inlet-mfl for "Mass Flow" boundary condition or bc-inlet-vel for "Velocity Inlet".
4. Similarly, use bc-outlet-spr for "Static Pressure", bc-outlet-ofl for outflow...
5. Walls with specified heat flux of heat transfer coefficient should also be named appropriately such as wf-htc- or wf-hfx- or wf-tmp-
7. Periodic boundaries can be named as prd-trn- or prd-rot-

#### Transient Table as Boundary Condition in FLUENT

A time-history data such as inlet temperature as function of time or mass flow rate as function of time can be specified either as an UDF (cases where analytical equations can be derived) or as a data table.

The format of the standard transient profile file:

```(
(inletvel transdata 3 0)
(time  1 2 3 )
(invel 6 5 4 )
)```
Here '3' denotes the number of data points (number of rows in tabular format). '0' indicated that data is not time-periodic, use '1' for a time-periodic dataset. 'invel' shall appear as drop-down in boundary condition panel. Note:
• All quantities, including coordinate values, must be specified in SI units.
• Profile names (e.g. 'inletvel' in this example) must have all lowercase letters
• Data can be either in column or row format.

The data set used above can also be specified as described below:

```(
(inletvel transdata 3 0)
(time
1
2
3
)
(invel
6
5
4
)
)```

Tabular Transient Profiles

The format of the tabular transient profile file is as follows. Use TUI command "file read-transient-table fileTr.dat" to upload the data into the solver. The options "inletvel time" and "inletvel invel" shall appear in the drop-down menu of boundary condition panels.
```inletvel transdata 3 0
time   invel
1     6
2     5
3     4```

##### Interpolation of Results
Many a times we need to interpolate the results from previous simulations into a new simulation, such as when a new mesh is generated due to refinement or coarsening. Sometimes results may be available from simulations carried out in other software (such as FLUENT or OpenFOAM) and the field variables need to be read into a new software (STAR-CCM+). Most of the commercial program provide an options to import and export data from and into CGNS and/or CSV format. The data in this format can be used to exchange result data from one program to the other.

Data Import into ANSYS FLUENT:

Data Export into CSV Format: ANSYS FLUENT:

The header in CSV files for FLUENT and STAR-CCM+ uses different variable names. In STAR-CCM+ variables has to be specified with appropriate units: e.g. "Absolute Pressure (Pa)", "Velocity Magnitude (m/s)", "Velocity[i] (m/s)", "Velocity[j] (m/s)", "Velocity[k] (m/s)", "X (m)", "Y (m)", "Z (m)"... Note the space between variables and unit. The x-component of velocity is accessed by Velocity[i]. Data Mapper: STAR-CCM+:

##### TUI in ANSYS FLUENT
Over the different versions, while new features have been added, the name of some of command have been altered and many have been removed / consolidated. For example, 'grid' has been replaced with 'mesh', /display/ set/ colors/ background white" has been replaced by "/display/ set/ colors/ graphics-theme-color white".