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기술자료 Strand7 Modelling potential fluid flow in Strand7

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작성자 씨앤지소프텍
댓글 0건 조회 6,715회 작성일 21-01-27 15:49

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Modelling potential fluid flow in Strand7

Strand7 has the capability to model simple fluid flow assuming irrotational potential flow. There are two basic formulations used in modelling potential flow, these are based on velocity potential (f) or stream function (j). Either of these formulations may be used in Strand7 and each can provide different information about the flow. Strand7 does not have a dedicated fluid modelling capability and hence the analysis of potential flow is done by analogy using the linear static and steady state heat solvers. Potential fluid flow can be modelled in 2D and 3D using the plate and brick elements.

The limitations of potential flow analysis must be emphasised. Potential flow analysis is only useful where viscous effects are negligible and where the flow is inertia dominated. The effects of the boundary layer cannot be considered. The flow is assumed to be irrotational and as such turbulence does not exist in this idealised representation. The method is generally limited to modelling flow in the presence of a favourable pressure gradient where separation and transition to turbulent flow are unlikely to occur.

Normally when modelling fluid flow using finite element techniques a section of the flow field is isolated and modelled with appropriate boundary conditions. These boundary conditions define the flow in and out of the section across part of the boundary. Other edges of the mesh will represent walls where the flow is tangent to the edge only (i.e. no normal flow across the edge). The extent of the mesh required for internal flows is easily defined; a finite element model of the internal volume is required with appropriate conditions defining the flow into and out of the pipe or channel. External flows are more difficult. Basically to model an external flow the fluid around the outside of the body must be modelled. The mesh should be extended far enough away from the body so that it includes the entire area of the flow influenced by the body. The boundaries of the mesh should extend out to the point where free stream conditions exist. If this is not done then the flow field modelled will not be infinite and errors will result.

Velocity Potential Formulation

The velocity potential formulation is the most practical way to model most fluid problems because of the relative simplicity of the boundary conditions. The boundary conditions are easy to apply and are intuitive. In addition, the results from this method are in general more useful than the results from the stream function method.

The velocity potential method is implemented in the linear steady state heat solver by an analogy. This is as follows:

Temperature (T) = Velocity Potential f
Flux (q) = Fluid Velocity (V)

In the heat properties the conductivity of the material should be set to K= 1.

The velocities are readily calculated from the velocity potential as follows:

X Velocity Component:
u = fx i.e. the derivative of velocity potential with respect to X.

Y Velocity Component:
v = fy i.e. the derivative of velocity potential with respect to Y.

Since the velocity potential is actually temperature in the analysis these derivatives correspond to the flux (i.e. fx = qx = dT/dX etc) and are obtainable from the solution.

The simplicity of this method lies in the ease of application of the boundary conditions. Both first order and second order boundary conditions may be assigned. With a first order condition the value of velocity potential is defined using a nodal temperature. By far the most useful of the two types of boundary conditions is the second order condition. This allows the velocity at a boundary to be defined. In the analysis of any fluid problem using finite elements the fluid itself is modelled. The edges of the mesh represent the walls of the duct etc that constrain the flow. The basic condition at any edge of this type is that flow is tangential to the edge - there is no fluid flow in the direction normal to the edge. Thus in this case the condition that we apply is that the velocity (or flux) normal to the edge of the mesh is zero. This is in fact the default condition for any free edge of a mesh in the heat solver and thus we simply do nothing to enforce the tangency flow condition on any free edge.

In most cases the boundary conditions used in this type of analysis are simply to define velocity and distribution of the flow in and out of the section.

If the boundary conditions are defined entirely by second order conditions (i.e. velocity) then the problem will be singular. At least one node must have a value assigned for the velocity potential (f). This can be any arbitrary value since the absolute value of potential has no real significance. It is the difference between the potential at two points in the flow field that is of importance. The derivatives of (f) are also of interest since these are the velocity. The derivatives are however independent of the magnitude of velocity potential.


The potential method provides results for velocity potential (Temperature) and flow velocity (flux). The velocity potential is best displayed as a line contour which is a grid orthogonal to the streamlines. The velocity (flux) is available in component or RMS forms. The RMS value will be the magnitude of the resultant component of velocity. The pressure at any point in the flow field or on surfaces can be readily calculated using the following equation:

Dynamic Pressure = q = 0.5 * r * V2

the coefficient of pressure Cp can be determined from the following:

Cp = 1 - (V/V¥)2

where V¥ is the free stream velocity.

Following are two examples of the use of the velocity potential method.

2D Plate Model

This example illustrates the use of Strand7 for modelling external flow around an infinitely long cylinder. A 2 dimensional slice of the flow field through the cylinder is modelled using 8 node plate elements. This mesh is extended far away from the cylinder so that there will be approximately free stream flow near the edges of the mesh. Only one quarter of the flow field is modelled due to symmetry of the flow. The Strand7 mesh is shown in the following figure:

Flow_1.gif

The boundary conditions applied are as follows:

Flow_2.gif

The fluid is assumed to be flowing in the positive X direction entering the mesh at the left hand edge. Along this edge the velocity of the free stream is defined using and edge flux (i.e. FluxX). A unit velocity is assumed in the model. When marking this condition onto the edge of the elements the sign of the flux is actually negative due to the sign convention assumed for plate elements: FluxX = -1 is applied to the edges of all the elements along the edge. This assumes that the velocity distribution is constant over the depth of the flow.

The lower edge of the mesh is the stagnation streamline. This is the vertical plane of symmetry and thus there is no normal flow along this edge - the flow is tangential. The boundary condition assigned to this edge is zero velocity in the normal direction or FluxY = 0 (i.e. Vy = 0). This is the default condition for a free edge and thus no condition needs to be applied to the edge.

Since the mesh has been extended a log way from the cylinder the flow at the upper edge is assumed to be free stream. Free stream velocity is parallel to the X direction thus there is no normal flow across this edge. The edge condition applied to this edge is Vy = 0 or FluxY = 0. As for the lower edge this is the default condition and no edge conditions need be applied to the mesh. Similarly the condition on the cylinder is that there is no normal flow; the fluid flows tangential to the surface. This is the default condition. The edge fluxes are actually applied in the local coordinate system of the plate elements. For each free element edge around the hole the local normal flow will be zero.

The condition on the vertical plane of symmetry is that the vertical fluid velocity is zero (i.e. Vy = FluxY = fy = 0). To enforce this, we set T=0 along the whole edge.

If a half model of the flow field is used, like the model used in the 2D stream function example below, the velocity of the fluid leaving the mesh also needs to be defined. In this case FluxX = 1 would also be applied to the right hand vertical edge.

The heat properties used are simply Kxx = Kyy = 1. The model was run using the linear steady state heat solver.

A plot of the velocity potential is shown below.

Flow_3.gif

The following figure shows a contour plot of the fluid velocity. Notice that velocity is zero at the stagnation point and a maximum (2xfree stream) at the point of maximum thickness.

Flow_4.gif

The results were compared with a theoretical potential flow solution for flow around a cylinder. This solution is obtained by superimposing the flow field from a source and a doublet. The tangential surface velocity of the flow around a 90 degree segment from the leading stagnation point (q = 0) to the maximum thickness (q = 90) is calculated at each of the nodal points in the mesh. This is compared with the tangential velocity (FluxRMS). The Flux RMS will be the tangential velocity in this case since there is no normal flow velocity at the surface of the cylinder. The following graph shows the Theoretical and Strand7 solution.

(Ref: Foundations of Aerodynamics by Kuethe and Chow)

Flow_5.gif

3D Brick Model

A similar example is run using brick elements. In this example we are modelling flow down a square duct with a cylindrical obstruction. The heat properties are idential to those used for the model above. The only difference is that as the flow is internal the extent of the mesh is defined by the walls of the duct.
Flow_6.gif

The boundary conditions are shown in the following figure:

Flow_7.gif


The following figure shows a contour plot of the velocity potential (f).

Flow_8.gif

The absolute velocities (FluxRMS) are shown in the following contour plot. Note that the surface velocities are slightly higher in this case than in the previous analysis of the external flow. This is because of the flow field is constrained by the duct and is no longer infinite. The velocities are approximately 20% higher than for the external case. If we used the above mesh to model the external case then the results would be 20% in error due to the semi-infinite nature of the mesh.

Flow_9.gif

Note that the contour plots of velocity potential and velocity have the same basic form as those for the 2D plate representation. The magnitude is however different due to the effect of the duct.


Stream Function Formulation

The stream function formulation can be useful as the results can produce plots of streamlines which are not available from the simpler potential method. Unfortunately the stream function method is limited to modelling simple structures because of the difficulty of defining the boundary conditions.

The stream function approach uses an analogy based on the linear static solver. In this case we assume that the only variable is the DX displacement. The other displacements are eliminated from the solution by the application of a global freedom condition. The DX displacement then becomes the stream function (y).

The concept of stream function is a little abstract. We can define a streamline as a curve whose tangent at every point coincides with the direction of the velocity vector i.e. there is no normal flow across a streamline. For incompressible flow the stream function is a measure of the volume of fluid that is flowing between two streamlines. This will be a constant at any point along two adjacent streamlines. The following figure illustrates this. The lines cd and ab are streamlines and thus no fluid can cross these lines. If we assume incompressible flow the density of the fluid within the boundary efgh does not vary with time and thus the same volume of fluid must cross ef per unit time as that crossing hg or any other path connecting the two streamlines.

Flow_10.gif
The stream function defines the volume of fluid per second passing between an arbitrary base streamline and some other streamline i.e. the amount of fluid flowing across the line ef or hg. It is normal to assume a reference streamline where an arbitrary value of flow rate (stream function) is assigned. For convenience this is normally assigned Y = 0. All other streamlines can then be identified by assigning them a value of stream function that is a measure of the volume of fluid flowing between the reference streamline and the streamline of interest.

The use of stream function is only practical for 2D flows and 3D axisymmetric flows. The stream function does exist in general 3D flow however the complexity of such flows will mean that it is virtually impossible to specify the appropriate boundary conditions to solve practical problems.

Velocities may be obtained from the stream function as follows:

X Velocity Component: u = Yy i.e. the derivative of stream function with respect to Y.
Y Velocity Component: v = Yx i.e. the derivative of stream function with respect to X.

The difficulty with the stream function approach is in the definition of the boundary conditions. It is not possible to define velocities at boundaries since the linear static solver has no mechanism to define the derivative of displacement as a boundary condition. The boundary conditions must be defined in terms of absolute values of stream function. Since the stream function corresponds to the X displacement the boundary conditions are set by enforcing particular values of DX along the boundaries. In most cases this will need to be done by the use of constraint equations. In some special cases where the stream function is zero along and edge freedom conditions may be used.

The output from the stream function is somewhat limited. The only data that can be output is the DX displacement or the stream function. This is best displayed as a line contour which will yield a plot of streamlines superimposed on the mesh. These is no mechanism to output the velocity since the derivatives of the X displacement are not available.

The use of the stream function is illustrated by the following two examples:

2D Plate Model

The problem considered is the same as that used for the 3D velocity potential case. A 2 dimensional slice of a section through the duct and cylindrical obstruction is modelled using 8 node plate elements. For this example a half model is used. In the case of stream function it is not possible to define a symmetry boundary condition. This requires the definition of the rate of change of stream function (yx) or since we are using the linear static solution with the DX displacement representing stream function, the rate of change of DX displacement must be defined. This is not possible. The half model used is shown in the following figure:

Flow_11.gif

The boundary conditions applied are as follows:

Flow_12.gif

Basically the stream function has been set to zero along the stagnation stream line where the velocity will be zero. On the outer surface of the duct the stream function has been set to Y = 1. This means that the volume of fluid flowing per unit time between the stagnation streamline and the outer surface of the duct is 1 m3/sec / unit depth (assuming that the units used to build the finite element mesh are metres).

The material properties used are E = 1, n = 0. The solution type is Plane Stress with a membrane thickness of T = 1.

The linear static solver was used for the solution.

The only useful result from available from this analysis is a contour plot of the stream function. This is shown in the following figure.
 
Flow_13.gif


3D Brick Model

The problem used here is the same as that considered for the 3D velocity potential example. Although this example is carried out in 3D it is really a 2D problem, since there is no three dimensional flow (i.e. any particle will move in a 2D plane only).

The same mesh is used. The material properties are the same as for the 2D stream function example.

The boundary conditions applied to the model are as follows:

Flow_14.gif


The contour plot obtained for stream function is shown in the following figure:

Flow_15.gif

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