2D Representation of Inviscid Air Flow over a Sphere

Abstract

This lab is a 2D representation of inviscid air flow over a sphere using the analogy of constant voltage lines on conductive paper with stream lines and potential lines. Lines were traced at several voltages measured along the paper. These lines, depending on how the paper was oriented in relationship to the current flow, corresponded to streamlines and velocity potential lines. Streamlines and velocity potential lines can dramatically simplify the analysis of a fluid flow. The results of the experiment were valid for inviscid fluid flows only and therefore did not exhibit rotational or boundary layer conditions.

Table of Contents

Abstract..………………………………………………………………………………..…ii

Table of Contents…………………………………………………………………………iii

Nomenclature………………………………………………………………………………iv

List of Figures……………………………………………………………….…………….v

List of Tables……………………………………………………………………….…….vi

Introduction…………………………………………………………………………………1

Discussion of Relevant Theory………………………………………………..……………2

Methods…………………………………………………………………………..…………6

Results………………….….………………………………………………………………7

Discussion…………………………………………………………………………………8

Conclusions………………………………………………………………………………..10

References……….…………………………………………………………………………11

Figures and Tables………..…………………………………………….………………….12

Appendices……………………………………………………………………………….14

Nomenclature

Arbitrary Constant…………………………………………………………………………………c

Direction Components of Velocity……………………………..………………………………….…u,v,w

Coordinate System Variables…………………………………………………………………. x,y,z

Voltage…………………………………………………………………………………………….V

Velocity………………………………………………………………………………………….. V

Gradient………………………………………………………………………………………….. $\mathbf{\nabla }\mathit{ }$

Laplace Operator………………………………………………………………………………… ${\mathbf{\nabla }}^{\mathit{2}}$

Stream Function………………………………………………………………………………….. $\mathrm{\Psi }$

Velocity Potential Function……………………………………………………………………… Φ

Vorticity………………………………………………………………………………………….. ξ

Angular Velocity………………………………………………………………………………… ω

List of Figures

Figure 1: Anderson Figure 2.28 Streamlines………………………….…………………12

Figure 2: White Figure 1-4 Flow Past a Circular Cylinder……………….……..….……12

Figure 3: Marking Points of Constant Voltage…………………………………………..13

Figure 4: Experimental Drawing………………………………………………..….……13

List of Tables

No Tables Used in this Experiment

Introduction

 This experiment uses the analogy between the aerodynamic stream function, velocity potential, and voltage in 2D to represent inviscid air flow around a circle. According to Anderson, “A streamline is a curve whose tangent at any point is in the direction of the velocity vector at that point” [1], as shown in figure 1.   Streamlines are important to aerodynamicists because they are a visualization of airflow. Visualization is key to understanding and developing proper dynamic models of flowing gasses. Streamlines are designated by setting the stream function ( $\mathrm{\Psi }$

) equal to a constant. When differentiated, the stream function gives the flow-field velocities. This is a very important characteristic of the stream functions.

In a similar manner, the derivatives of the velocity potential give flow-field velocities. Unlike the stream function, the velocity potential is in the same direction as the flow-field velocities, can be applied to 3D flows, and can be applied to irrotational flows only. The velocity potential is important because it allows aerodynamicists to drastically simplify flow fields.

By running a voltage around a circular shape on a piece of conductive paper an illustration of velocity potential and streamlines of an inviscid flow is created. This is a very simplified example of airflow, but with real world applications. This experiment is meant to help the aerodynamicist visualize the stream and potential functions and understand how using an analogy can be beneficial in science and engineering.

Discussion of Relevant Theory

Visualization of fluid flow over a solid object is a key component of dynamic modeling and aerodynamics. “Inviscid flows do not truly exist in nature; however, there are many practical aerodynamic flows (more than you would think) where the influence of transport phenomena is small, and we can model the flow as being inviscid” states Anderson [2]. 

The analogous affect in this lab is credible due to the fact that the stream function, velocity potential function and voltage (V) all simultaneously appease the Laplace Equation.

${\mathbf{\nabla }}^{\mathit{2}}\mathrm{\Psi }=0,\mathrm{ }{\mathbf{\nabla }}^{\mathit{2}}\mathrm{\Phi }=0,\mathrm{ }{\mathbf{\nabla }}^{\mathit{2}}\mathrm{V}=0$

For this to hold true in aerodynamics, there must be a complete absence of shear forces in the flow field. Those shear forces include sources, sinks and rotation. Electrostatically this analogy is satisfied if there is no net charge present on the conducting paper.

The stream function ( $\mathrm{\Psi }$

) and velocity potential (Φ) are two mathematical constructs that are used to describe a flow and its interaction with a solid object. One can use the equation for the incompressible stream function to give the equation for a streamline by setting the stream function equal to a constant, as shown here 

$\mathrm{\Psi }(x,y$

) = c

The velocity is acquired by differentiating ψ in as shown below

${\mathbf{V}}_u=\frac{\partial \mathrm{\Psi }}{\partial y}$

${\mathbf{V}}_v=–\frac{\partial \mathrm{\Psi }}{\partial x}$

The boundary condition of this function is that the derivative parallel to the solid surface is zero. The comparison with the voltage in the experiment is that the voltage is constant along a perfect conductor and therefore is analogous with the stream function. This is to say that the fluid cannot flow into the solid and voltage cannot flow into a perfect conductor. Setting the stream function equal to several different constants allows for the sketching of the fluid flow around a solid. This is important because it allows aerodynamicists to be able to represent two velocity components with one variable.

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Before describing the velocity potential, it is important to establish that this experiment is valid only for an inviscid flow. As stated by White, this lab “illustrates a characteristic of inviscid flow without a free surface or ‘deadwater’ region: There are no parameters such as Reynolds number and no dependence upon physical properties… the integrated surface pressure force in the streamwise direction, the cylinder drag, is zero. This is an example of the d’Alembert paradox for inviscid flow past immersed bodies” [3]. This being true allows for a vacancy of viscosity in the lab’s analogy.

Viscosity gives rise to two major factors in real flow: boundary layer thickening and vorticity effects on the trailing edge of the solid object. The boundary layer thickening effect can be simulated in this analogy by simply drawing the shape of the circle thicker than normal to account for the additional layer. The vorticity effects are created by viscosity, rotation, and distortion in a fluid. These effects cause circulation of the airflow around the solid body.  The equation for vorticity (ξ) is given by

$\xi =2\omega$

Where ω is the angular velocity of the solid body. This gives us a relationship between vorticity and the velocity field by the equation

$\xi =\mathrm{ }$

  $\mathbf{\nabla }\mathit{ }\times$

V

The above equation says that the curl of the velocity field is equal to the vorticity. This relationship allows for a distinction between rotational and irrotational flows. If the flow is irrotational, ξ = 0. Irrotational flows are much simpler to analyze than rotational flows. The velocity potential (Φ) can now be used to mathematically describe the flow. The equation for the velocity potential is expressed by

$\mathbf{V}\mathbf{=}\mathbf{ }\mathbf{\nabla }\mathit{ }$

Φ

This equation says that for an irrotational flow there is a scalar function Φ such that the velocity is given by the gradient of Φ. The velocity function can then be differentiated to yield

$u= \frac{\partial \mathrm{\Phi }}{\partial x},\mathit{ v}= \frac{\partial \mathrm{\Phi }}{\partial y},\mathit{ w}= \frac{\partial \mathrm{\Phi }}{\partial z}$

These equations show the characteristics stated earlier that the velocity potential is valid for 3D flows, it is in the same direction as the velocity vectors, and it applies to irrotational flows only. Furthermore, the equations reveal the orthogonal nature of the potential function to the stream function.

$u= \frac{\partial \mathrm{\Phi }}{\partial x}= \frac{\partial \mathrm{\Psi }}{\partial y}$

Except at the stagnation points, where the velocity vanishes, the streamlines and potential lines are orthogonal to each other because the ratios of the partial differentiations are negative reciprocals.

Now the flow can be expressed as a one equation dealing with one unknown which is a dramatic simplification of the fluid flow. The boundary condition for the velocity potential is that the derivative normal to a solid body is zero, so the lines must approach perpendicularly to the surface body. Likewise, the voltage lines must be perpendicular to the outline of the perfect insulator in the experiment. This is equivalent, as with the stream function, of the flow not going through the solid.

Assuming there is no net charge on the conducting paper, flow around the circle should resemble that as shown in figure 2. Symmetric streamlines and velocity potential lines will remain continuous and reflect across the stagnation voltage line through the center of the circle. No viscosity shall be present, and no flow separation shall be experienced.

Methods

This experiment used a 23-inch square board with electrodes on two opposite sides. The electrodes were connected to a DC power source.  A piece of conducting paper with a high resistance was placed on the board and a voltage drop of 10 volts was established through the paper from the electrodes. An outline of a circle was drawn and filled in with very conductive nickel paint. The lines of constant voltage ran parallel to the two electrodes, so the circle had to be oriented such that it too is parallel. After allowing the paint to dry, a digital voltmeter was used to measure voltages. As pictured in Figure 3, several (5 – 10) measurements were taken across the board and the lines of constant voltage were traced with a pen. These lines were analogous to the streamlines.

 After turning off the power supply, the paper was removed, and the outline of the circle was cut out of the paper leaving a hole in the same shape. The paper was replaced on the board but rotated 90 degrees from the original orientation. Repeating the previous process, constant voltage lines were once again measured and traced. These lines were representative of the potential function lines. The area was then cleaned up and everything returned to its place.

Results

The experiment showed the constant voltage lines flowing around the circle in a manner that is analogous to inviscid air flow. For the sake of time, only 5 lines were drawn for the potential lines, but this was sufficient to represent the analogy effectively. An image of the lab test results is shown in figure 4.  The circular body was oriented such that the flow was moving from left to right, with the left-hand side of the circle being the leading edge. Measurements for the streamlines were taken at 3.60, 4.25, 4.85, 6.00, 6.50, and 7.00 volts. The flow velocity should always be tangential to these lines, which is reflected in the figure. The closer the lines are to each other around the body is where the air flow would be moving faster. Measurements for the potential lines were taken at 2.5, 3.65, 4.5, 5.45, and 6.70 volts. The velocity of the flow should always be perpendicular to these lines, which is once again reflected in the figure.

Discussion

Inviscid flow is attained as the Reynolds number approaches infinity in theory. In the case of very high but a finite Reynolds number, the flow can be practically assumed to be inviscid. Anderson notes that “For such flows, the influence of friction, thermal conduction, and diffusion is limited to the boundary layer, and the remainder of the flow outside is essentially inviscid…For flows over slender bodies, inviscid theory adequately predicts the pressure distribution and lift on the body and gives a valid representation of the streamlines and flow field away from the body.” [2].

The vorticity condition was not seen acting along the trailing edge of the circle. In figure 4 it is observable that there was no circulation of the constant voltage lines (streamlines) around the trailing edge. This was due to the lack of viscosity and rotation as discussed in the relevant theory section. If the flow was viscid there would also have been a boundary layer thickness on the body, separation of the boundary layer at the trailing edge, and turbulence in the flow. None of these conditions were present in this experiment. Since there was no friction in the flow there was also no lift or drag on the circle as evident during the experiment since the paper had no forces acting upon it.

Upon inspection of figure 4, the streamlines appear approximately symmetric across the stagnation value. Minute fluctuation in the results are most likely the result of human error. These errors could include not painting a symmetric circle causing an imperfect conductor, or unsteadiness while tracing the points of constant voltage. Fluctuations in the potential lines are suspect to similar human errors. Cutting out an unsymmetrical circle leading to an imperfect insulator, and again errors while tracing the points of constant voltage.

Overall the experiment worked well to convey the analogy of electric current to inviscid air flow.

Conclusion

Overall this lab was about visualization of air flow and understanding how useful analogies can be. The results of the lab turned out as expected. The streamlines and potential lines were clearly visible flowing around the circle in a manner similar to inviscid airflow. The absence of viscosity meant that there was no boundary layer thickness or separation. It also meant there was no vorticity. This is a major difference from real viscous flow, but inviscid flow is easily analyzed and easily produced.

References

[1] Anderson, J. D. Jr., “Fundamentals of Aerodynamics”, 5^th ed., McGraw Hill, New York, 2011, pp.  158-185.

[2] Anderson, J. D. Jr., “Fundamentals of Aerodynamics”, 5^th ed., McGraw Hill, New York, 2011, pp.  62-64.

[3] White, Frank M., “Viscous Fluid Flow”, 3^rd ed., McGraw Hill, New York, 2006, pp. 4-12.