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Table of Contents
  1. Incompressible, inviscid flow
    1. Basic flow elements
      1. Uniform flow
      2. Source
      3. Sink
    2. Superposition of elements
      1. Rankine half-body
        1. Oval rankine half-body
      2. Doublet
      3. Line Vortex
      4. Flow around cylinder
      5. Flow around cylinder with circulation
    3. Method of images
      1. Source above wall
      2. Advection of vortices
  2. Compressible flow
    1. Mach number relations
    2. Shockwaves
    3. Nozzles
    4. Fanno flow
    5. Rayleigh flow
  3. Open channel flow
    1. Surface waves
    2. Energy in OCF
    3. Gradually varied flow
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TEP4135: Fluid Mechanics 2

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Incompressible, inviscid flow

Basic flow elements

Uniform flow

$$\phi = u_{\infty}x + v_{\infty}y $$ $$\psi = u_{\infty}x - v_{\infty}y $$

Source

$$\phi = \frac{q}{2\pi} ln \space r$$ $$\psi = \frac{q}{2\pi} \theta $$

Sink

$$\phi = \frac{-q}{2\pi} ln \space r$$ $$\psi = \frac{-q}{2\pi} \theta $$

Superposition of elements

Superposition combines basic flow elements to create more advanced flow fields. The combination is linear, and the final expressions of the field will be the sum of functions representing the different basic elements

Rankine half-body

A Rankine half-body is a source added to a uniform flow.

$$ \phi = u_{\infty}x + \frac{-q}{2\pi} ln \space r$$ $$\psi = u_{\infty}y\frac{-q}{2\pi} \theta $$

The stagnation point $s$ of a Rankine half-body is given by $$ s = \frac{q}{2\pi u_{\infty}}$$ $$ q = 2\pi s u_{\infty}$$ The streamline which passes thorugh he stagnation point, passes along $y = 0$ when $\theta = \pi$, as given by $$ \psi = u_{\infty}(y + s\theta \\ \Rightarrow \psi_s = u_{\infty}s\pi \\ = \frac{q}{2}$$

Pressure distribution is given by $$ C_p = - \left(\frac{2s \space cos \theta}{r} + \frac{s^2}{r^2} \right)$$

Oval rankine half-body

A Rankine oval is a sink and a source in a uniform flow, spaced apart from eachother. The stagnation streamline will then create a boundry between the external streamlines of the uniform flow, and the internal streamlines that starts in the source and ends in the sink.

Doublet

A doublet is a source added on top of a sink, both equal in strength. Streamlines will then circulate from the center of the sink/source outwards, make a loop, and then end in the center.

$$ \phi = \frac{q}{2\pi}\frac{cos \theta}{r}$$

The strength of a doublet can be calculated by $$ B = \frac{q \space a}{2\pi}$$

Note that this flow elements has directionality, which means it is sensitive to the values provided.

Line Vortex

A line vortex causes streamlines to move in a circular fashion around a certain point in the field. The flow of a line vortex has no radial movement. $$ \phi = \frac{\Gamma}{2\pi r}$$ $$ \psi =\frac{\Gamma}{2\pi r} \space ln \space r$$ where $\Gamma$ is a constant representing the circulation.

Flow around cylinder

A flow around a cylinder is built up of doublet placed in a uniform flow. $$ \phi = u_{\infty}r \space sin\theta - \frac{B}{r}sin\theta$$

The pressure distrubution is given by $$C_p = 1-4 \space sin^2 \theta$$

This gives a symmetric pressure distrubution, resulting in no net forces (which is unphysical).

In reality, drag and lift of a cylinder is given by $$ C_D = \frac{F_x}{\frac{1}{2} \rho u^2_{\infty} D^2} \\ C_L = \frac{F_y}{\frac{1}{2} \rho u^2_{\infty} D^2}$$

The drag is caused by skin friction (viscous effects) and pressure drag. A real cylinder in a flow will have a messy wake.

Flow around cylinder with circulation

A flow around a cyliner with circulation is modeled by adding a line vortex on top of the flow around of a cylinder.

Method of images

Method of images is used to represent a solid boundry in a flow field by adding flow elements whose streamlines acts as a solid boundry.

Source above wall

Advection of vortices

Compressible flow

Mach number relations

Shockwaves

Nozzles

Fanno flow

Rayleigh flow

Open channel flow

Surface waves

Energy in OCF

Gradually varied flow

Written by

tajoon
Last updated: Tue, 15 Dec 2020 21:31:14 +0100 .
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