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Point Velocity Measurements P M V Subbarao Professor Mechanical Engineering Department Velocity Distribution is a Fundamental Symptom of Solid –Fluid Interactions…. Point Velocity Measurement • Pitot Probe Anemometry : Potential Flow Theory . • Thermal Anemometry : Newton’s Law of Cooling. • Laser Anemometry: Doppler Theory. POTENTIAL FLOW THEORY • Ideal flow past any unknown object can be represented as a complex potential. • In particular we define the complex potential W i In the complex (Argand-Gauss) plane every point is associated with a complex number i z x iy re In general we can then write W z i z f z Now, if the function f is analytic, this implies that it is also differentiable, meaning that the limit df f z z f z lim dz z 0 z dW u iv dz so that the derivative of the complex potential W in the complex z plane gives the complex conjugate of the velocity. Thus, knowledge of the complex potential as a complex function of z leads to the velocity field through a simple derivative. ELEMENTARY IRROTATIONAL PLANE FLOWS • The uniform flow • The source and the sink THE UNIFORM FLOW The first and simplest example is that of a uniform flow with velocity U directed along the x axis. In this case the complex potential is W i Uz and the streamlines are all parallel to the velocity direction (which is the x axis). Equi-potential lines are obviously parallel to the y axis. THE SOURCE OR SINK • source (or sink), the complex potential of which is m W i ln z 2 • This is a pure radial flow, in which all the streamlines converge at the origin, where there is a singularity due to the fact that continuity can not be satisfied. • At the origin there is a source, m > 0 or sink, m < 0 of fluid. • Traversing any closed line that does not include the origin, the mass flux (and then the discharge) is always zero. • On the contrary, following any closed line that includes the origin the discharge is always nonzero and equal to m. Iso lines Iso lines The flow field is uniquely determined upon deriving the complex potential W with respect to z. m W i ln z 2 Stream and source: Rankine half-body • It is the superposition of a uniform stream of constant speed U and a source of strength m. 2D Rankine half-body: m W i Uz ln z 2 m W i Uz ln z 2 i Z x iy re x r cos y r sin y tan 1 x i W i Ure m 2 ln re i ln r m m i Ux iUy i 2 2 m ln r iUy m i Ux 2 2 m Uy 2 Shape of Zero Value Stream line m 0 Ur sin 2 2D Rankine half-body: y m 1 y 0 Uy tan 2 x 3D Rankine half-body: m Ux 4 x 2 y 2 z 2 Pitot Probe Anemometry : Henri Pitot in 1732 • Theory • A constant-density fluid flowing steadily without friction through the simple device. • No heat being added and no shaft work being produced by the fluid. • A simple expression can be developed to describe this flow: u12 u2 2 p1 gz1 p2 gz2 2 2 Apply Bernoulli’s equation along the central streamline from a point upstream where the velocity is u1 and the pressure p1 to the stagnation point of the blunt body where the velocity is zero, u2 = 0. Also z1 = z2. 1 2 This increase in pressure which bring the fluid to rest is called the dynamic pressure. 2 u p2 p1 1 2 Dynamic pressure = 2 or converting this to head p1 u p2 1 g 2 g g Dynamic head = The total pressure is know as the stagnation pressure (or total pressure) 2 u Stagnation pressure = p2 p1 1 2 2 p1 u or in terms of head Stagnation head = 1 g 2 g The blunt body stopping the fluid does not have to be a solid. It could be a static column of fluid. Two piezometers, one as normal and one as a Pitot tube within the pipe can be used in an arrangement to measure velocity of flow. Using the above theory, we have the equation for p2 , p2 p1 u12 u 2gh2 h1 2 We now have an expression for 2 velocity obtained from two pressure p1 u p2 1 measurements and the application of g 2 g g the Bernoulli equation. Pitot Static Tube • The necessity of two piezometers and thus two readings make this arrangement is a little awkward. • Connecting the piezometers to a manometer would simplify things but there are still two tubes. • The Pitot static tube combines the tubes and they can then be easily connected to a manometer. • A Pitot static tube is shown below. • The holes on the side of the tube connect to one side of a manometer and register the static head, (h1), while the central hole is connected to the other side of the manometer to register, as A Pitot-static tube before, the stagnation head (h2). Consider the pressures on the level of the centre line of the Pitot tube and using the theory of the manometer, p A p2 gX pB p1 g X h man gh p A pB X p2 gX p1 g X h man gh h B A 2 u We know that p2 pstag p1 1 2 2 p1 gh man p1 u 1 2 2 gh man u1 The Pitot/Pitot-static tubes give velocities at points in the flow. It does not give the overall discharge of the stream, which is often what is wanted. It also has the drawback that it is liable to block easily, particularly if there is significant debris in the flow. Compressible-Flow Pitot Tube For an ideal compressible flow coming to rest from finite velocity: 2 V h h0 2 For perfect gas : 2 V c pT c pT0 2 2 c p T0 T V 2 This process of ideal compressible flow coming to rest is regarded as isentropic process. Tds dh vdp 0 For perfect gas : c p dT vdp RT For ideal gas : c p dT dp p dT dp 1 T p p0 T0 1 p T Compressible-Flow Pitot Tube Subsonic pitot tube : A pitot tube in subsonic flow measures the local total pressure po together with a measurement of the static pressure p 2 p0 c p T0 T c pT 1 V 2 p p0 V 2 1 M V 1 p 2c pT RT 1 p0 1 2 1 2 p0 1 M M 2 1 p 2 1 p The pitot-static combination therefore constitutes a Mach meter . With M2 known, we can then also determine the dynamic pressure. 1 V 2 2 p0 M2 1 p 1 p The velocity can be determined from 1 2p p0 V2 1 1 p Supersonic pitot tube • A pitot probe in a supersonic stream will have a bow shock ahead of it. • This complicates the flow measurement, since the bow shock will cause a drop in the total pressure, from po1 to po2 , the latter being sensed by the pitot port. • It’s useful to note that the shock will also cause a drop in o, but ho will not change. • The pressures and Mach number immediately behind the shock are related by p02 1 2 1 1 M2 p2 2 Normal Shock Relations p02 1 2 1 1 M2 p2 2 ( 1)M1 2 2 2 • Mach number relation M2 2M1 1 2 • Static pressure jump relation p2 2M 1 2 1 p1 1 Mach Number Range