Consider An Irrotational And Inviscid Water Flow Crea 570326

Consider An Irrotational And Inviscid Flow Of Water Created Using A

Consider an irrotational and inviscid flow of water created using a superposition of a rankine oval and uniform flow. The sink-source pair of the oval is on the x–axis, with the source on the left (c = −1.5 m) and the sink on the right (c = +1.5 m). The strength of the sink and source is E = 40 m²/s. (a) Pick an arbitrary value for the magnitude of the uniform flow’s velocity. Then, derive and plot the contours for the stream and potential functions of this uniform flow as it flows over the Rankine oval. (b) Derive the velocity vectors (u and v). Then, find the velocity of the uniform flow to yield a Rankine oval with a total length 2l = 10 m. (c) Assume that the upstream pressure is zero. Calculate and plot the pressure distribution on the surface of the Rankine oval, for the uniform flow velocity found in part (b).

Paper For Above instruction

Understanding complex fluid flows, especially those involving potential flows superimposed with boundary shapes like the Rankine oval, is fundamental in fluid mechanics. The problem at hand involves analyzing an irrotational and inviscid flow created by superimposing a rankine oval with a uniform flow. This setup is classic in potential flow theory, where the flow is idealized as inviscid and irrotational, enabling the use of complex potential functions for analysis.

Superposition of the Rankine Oval and Uniform Flow

The superposition principle allows the creation of complex flow fields by adding simpler, known solutions. In this case, a Rankine oval—a streamline pattern generated by a source-sink pair—is combined with a uniform flow. The source at c = -1.5 m and the sink at c = 1.5 m serve to shape the flow around the oval, and their strengths, denoted by E = 40 m²/s, control the size of the oval.

Part (a): Defining the Potential and Stream Functions

Choosing an arbitrary magnitude for the uniform flow velocity, U, we can write the complex potential for the uniform flow as:

\[ \Phi_{U} = U x + i U y \]

with the corresponding stream function:

\[ \Psi_{U} = U y \]

superimposed onto the potential and stream functions for the source-sink pair and Rankine oval. The potential function for a source of strength E at position c is:

\[ \Phi_{\text{source}} = \frac{E}{2\pi} \ln|z - c| \]

and the sink has the same potential, with opposite sign.

To find the combined potential for the superposition, summing these potentials gives:

\[ \Phi(z) = U z + \frac{E}{2\pi} \left( \ln|z + 1.5| - \ln|z - 1.5| \right) \]

Plotting the Contours

Contours of the potential function \(\Phi\) represent the equipotential lines, enabling visualization of flow patterns. Similarly, the stream function \(\Psi\), derived as the harmonic conjugate of \(\Phi\), renders the streamlines of the flow. For a specific U value, the contours can be plotted numerically using these formulas. Such plots reveal the influence of the uniform flow on the Rankine oval's shape, illustrating how streamlines and potential lines distribute around the boundary.

Part (b): Deriving Velocity Vectors

The velocity components u and v are obtained from the potential function \(\Phi\) as:

\[ u = \frac{\partial \Phi}{\partial x} \quad \text{and} \quad v = \frac{\partial \Phi}{\partial y} \]

Applying these derivatives to the combined potential function yields the velocity field:

\[ u = U + \frac{E}{2\pi} \left( \frac{x + 1.5}{(x + 1.5)^2 + y^2} - \frac{x - 1.5}{(x - 1.5)^2 + y^2} \right) \]

\[ v = U y / r^2 \text{ terms from derivatives of the logs} \]

By analyzing the flow at the oval's surface, you can derive the value of U required to produce an oval of total length 2l = 10 m; specifically, U influences the extent of the flow separation and streamline shape.

Part (c): Pressure Distribution on the Surface

Assuming upstream pressure is zero (relative to atmospheric pressure), Bernoulli's equation applies along streamlines. The pressure at a point on the surface of the Rankine oval is:

\[ p = p_\infty + \frac{\rho}{2} (U_\infty^2 - |\mathbf{V}|^2) \]

where \(\mathbf{V} = (u, v)\) is the local velocity, and \(p_\infty = 0\). Computing \(|\mathbf{V}|\), the magnitude of the velocity vector, along the surface allows plotting of the pressure distribution. Regions of lower pressure indicate flow separation or potential cavitation zones, critical in engineering applications like vessel hull design and aerodynamics.

Conclusion

This combined potential flow approach effectively models complex flow features with analytically derived functions, making it instrumental for preliminary design and analysis in fluid mechanics. While idealized, these insights form the foundation for more nuanced numerical simulations accounting for viscosity, turbulence, and unsteady effects.

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