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Unit XI

Implementation of boundary conditions in CFD

The common boundary conditions used in CFD control volume method

are1. inlet

2. outlet

3. wall

4. prescribed pressure

5. symmetry

6. periodicity (or cyclic boundary condition)

The following assumptions are made for implementing boundary

conditions

i. the flow is always subsonic (M < 1), (ii) k- turbulence modelling

is used, (iii) the hybrid differencing method is used for discretisation and

(iv) the SIMPLE solution algorithm is applied.

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Inlet boundary conditions

The distribution of all flow variables are need to be specified at inlet

boundaries

Consider a grid arrangement in a flow field as shown in fig.

Figures 1, 2,3,4 shows the grid arrangement in the immediate vicinity of

an inlet for u- and v- momentum, pressure and scalor correction

equation cells. The flow direction is assumed to be broadly from the left

to the right in the diagrams.The grid extends outside the physical boundary and the nodes along the

line I= 1 are used to store the inlet values of flow variables. The

discretised equation is solved for the first internal cell, which is shaded

as shown in fig. 1.The diagrams also show the 'active' neighbours and cell faces which are

represented in the discretised equation for the shaded cell assuming

that hybrid differencing is used. For instance, in Figures the active

neighbour velocities are given by means of arrows and the active facepressures by solid dots.

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Fig.1u-velocity cell at the inlet boundary Fig.2 v-velocity cell at the inlet boundary

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Fig.3. Pressure correction fig.4 Scalar cell at inlet boundary

cell at inlet boundary

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The figures indicate that all links to neighbouring nodes remain active

for the first u-, v- and cell, so to accommodate the inlet boundary

condition for these variables it is unnecessary to make any modifications

to their discretised equations.

The pressure field obtained by solving the pressure correction equation

does not give absolute pressures. It is common practice to fix the

absolute pressure at one inlet node and set the pressure correction to

zero at that node. Having specified a reference value the absolute

pressure field inside the domain can be obtained.

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Outlet boundary conditions

Outlet boundary conditions may be used in conjunction with the inlet

boundary conditions. If the location of the outlet is selected far away

from geometrical disturbances the flow often reaches a fully developed

state where no change occurs in the flow direction. In such a region we

can place an outlet surface and state that the gradients of all variables

(except pressure) are zero in the flow direction.

Figures 1,2,3,4 show the grid arrangements near such an outlet

boundary.

If NI is the total number of nodes in the x-direction, the equations are

solved for cells up to I (or i) = NI 1. Before the relevant equations are

solved the values of flow variables at the next node (NI), just outside thedomain, are determined by extrapolation from the interior on the

assumption of zero gradient at the outlet plane.

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Fig.1 u-control volume at an outlet boundary fig.2 v-control volume at an outlet boundary

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Fig.3 Pressure correction cell at an outlet boundary Fig.4 Scalar cell at an outlet boundary

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For the v- and scalar equations this implies setting

Figures 2 and 4 show that all links are active for these

variables so their discretised equations can be solved as normal.

Calculation of u at the outlet plane i = NI by assuming a zero gradientgives

During the iteration cycles of the SIMPLE algorithm there is no

guarantee that these velocities will conserve mass over the

computational domain as a whole. To ensure that overall continuity is

satisfied the total mass flux going out of the domain (Mout) is first

computed by summing all the extrapolated outlet velocities. To make

the mass flux out equal to the mass flux Min coming into the domain all

the outlet velocity components are multiplied by the ratio

Min/Mout. Thus the outlet plane velocities with the continuitycorrection are given by

These values are subsequently used as the east neighbour velocities in

the discretised momentum equations for

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Wall boundary conditions

The wall is the most common boundary encountered in confined fluid

flow problems.Consider a solid wall parallel to the x-direction. Figures 1,2,3 illustrate

the grid details in the near wall regions for the u-velocity component

(parallel to the wall), for the v-velocity component (perpendicular to the

wall) and for scalar variables.

The no-slip condition (u = v =0) is the appropriate condition for the

velocity components at solid walls. The normal component of the

velocity can simply be set to zero at the boundary (j= 2) and the

discretised momentum equation at the next v-cell in the flow (j = 3) can

be evaluated without modification. Since the wall velocity is known it is

also unnecessary to perform a pressure correction here.

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fig 1 u-velocity cell at a wall boundary

fig.2

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Fig.3 Scalar cell at a wall boundary

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The constant pressure boundary condition

The constant pressure condition is used in situations where exact details

of the flow distribution are unknown but the boundary values of

pressure are known. Typical problems where this boundary condition isappropriate include external flows around objects, free surface flows,

buoyancy-driven flows such as natural ventilation and fires, and also

internal flows with multiple outlets.

In applying the fixed pressure boundary the pressure correction is set tozero at the nodes. The u-momentum equation is solved from i = 3 and

the v-momentum and other equations from I = 2 onwards. The main

outstanding problem is the unknown flow direction which is governed

by the conditions inside the calculation domain. The u- velocity

component across the domain boundary is generated as part of the

solution process by ensuring that continuity is satisfied at every cell. For

example, in fig 1, the values of ue and of vs and vn emerge from solving

the discretised u- and v- momentum equations inside the domain. Given

these values we can compute uw by insisting that mass is conserved for'-

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P-cell at an inlet boundary p'-cell at an outlet boundary

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This implementation of the boundary condition causes the p'-cell

nearest to the boundaries to act as a source or sink of mass. The process

is repeated for each pressure boundary cell. Other variables such as , T,

k and must be assigned inflow values where the flow direction is into

the domain. Where the flow is outwards their values just outside the

domain may be obtained by means of extrapolation

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Symmetry boundary condition

The conditions at a symmetry boundary are: (i) no flow across the

boundary and (ii) no scalar flux across the boundary. In the

implementation, normal velocities are set to zero at a symmetryboundary and the values of all other properties just outside the solution

domain (say I or i = 1) are equated to their values at the nearest node

just inside the domain (Ior i = 2):

In the discretised p'-equations the link with the symmetry boundary

side is cut by setting the appropriate coefficient to zero; no further

modifications are required.

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Periodic or cyclic boundary condition

Periodic or cyclic boundary conditions arise from a different type of

symmetry in a problem.

Consider for example swirling flow in a cylindrical furnace shown in

Figure. In the burner arrangement gaseous fuel is introduced through six

symmetrically placed holes and swirl air enters through the outer

annulus of the burner.

h bl b l d l d l l d ( ) b

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This problem can be solved in cylindrical polar co-ordinates (z, r, ) by

considering a 60 angular sector as shown in the diagram where k refers

to r-z planes in the -direction. The flow rotates in this direction, and

under the given conditions the flow entering the first k-plane of the

sector should be exactly the same as that leaving the last k-plane. This is

an example of cyclic symmetry. The pair of boundaries k = 1 and k =NK

are called periodic or cyclic boundaries.

To apply cyclic boundary conditions we need to set the flux of all flow

variables leaving the outlet cyclic boundary equal to the flux enteringthe inlet cyclic boundary. This is achieved by equating the values of each

variable at the nodes just upstream and downstream of the inlet plane

to the nodal values just upstream and downstream of the outlet plane.

For all variables except the velocity component across the inlet andoutlet planes (say w) we have

For the velocity component across the boundary we have