During World War II, Sir Geoffrey Taylor, a British fl uiddynamicist, used dimensional analysis to estimate theenergy released by an atomic bomb explosion. He assumedthat the energy released E , was a function of blast waveradius R , air density ρ, and time t . Arrange these variablesinto a single dimensionless group, which we may term theblast wave number .

During World War II, Sir Geoffrey Taylor, a British fl uiddynamicist, used dimensional analysis to estimate theenergy released by an atomic bomb explosion. He assumedthat the energy released E , was a function of blast waveradius R , air density ρ, and time t . Arrange these variablesinto a single dimensionless group, which we may term theblast wave number .

Principles of Heat Transfer (Activate Learning with these NEW titles from Engineering!)

8th Edition

ISBN:9781305387102

Author:Kreith, Frank; Manglik, Raj M.

Publisher:Kreith, Frank; Manglik, Raj M.

Chapter5: Analysis Of Convection Heat Transfer

Section: Chapter Questions

Problem 5.13P: 5.13 The torque due to the frictional resistance of the oil film between a rotating shaft and its...

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5.13 The torque due to the frictional resistance of the oil film between a rotating shaft and its bearing is found to be dependent on the force F normal to the shaft, the speed of rotation N of the shaft, the dynamic viscosity of the oil, and the shaft diameter D. Establish a correlation among these variables by using dimensional analysis.
During World War II, Sir Geoffrey Taylor, a British fluid dynamicist, used dimensional analysis to estimate theenergy released by an atomic bomb explosion. He assumed that the energy released E, was a function of blastwave radius R, air density ρ, and time t. Arrange these variables into single dimensionless group, which we mayterm the blast wave number.
When fluid in a pipe is accelerated linearly from rest, it begins as laminar flow andthen undergoes transition to turbulence at a time ttr which depends upon the pipe diameter D,fluid acceleration a, density ρ, and viscosity µ. Arrange this into a dimensionless relationbetween ttr and D. (Fluid Mechanics)
In the study of turbulent flow, turbulent viscous dissipation rate ? (rate of energy loss per unit mass) is known to be a function of length scale l and velocity scale u′ of the large-scale turbulent eddies. Using dimensional analysis (Buckingham pi and the method of repeating variables) and showing all of your work, generate an expression for ? as a function of l and u′.
Evaluate the use of dimensionless analysis using the Buckingham Pi Theorem for a given fluid flow system (D4) , where resistance tomotion ‘R’ for a sphere of diameter ‘D’ moving at constant velocity on the surface of a liquid is due to the density ‘ρ’ and the surfacewaves produced by the acceleration of gravity ‘g’. The dimensionless quantity linking these quantities is Ne= Function (Fr). To do thisyou must apply dimensional analysis to fluid flow system given in Figure 1 (P11). PICTURE IS ALSO ATTACHED
The Stokes number, St, used in particle dynamics studies,is a dimensionless combination of five variables: accelerationof gravity g , viscosity μ , density ρ , particle velocity U ,and particle diameter D . ( a ) If St is proportional to μand inversely proportional to g , find its form . ( b ) Showthat St is actually the quotient of two more traditionaldimensionless groups.
When a steady uniform stream flows over a circular cylinder, vortices are shed at a periodic rate. These are referred to as Kármán vortices. The frequency of vortex shedding få is defined by the free-stream speed V, fluid density p, fluid viscosity u, and cylinder diameter D. Use the Buckingham Pi method to show a dimensionless relationship for Kármán vortex shedding frequency is St = f (Re). Show all your work. V D
Suppose we know little about the strength of materials butare told that the bending stress σ in a beam is proportionalto the beam half-thickness y and also depends on thebending moment M and the beam area moment of inertiaI . We also learn that, for the particular case M = 2900in ∙ lbf, y = 1.5 in, and I = 0.4 in4 , the predicted stressis 75 MPa. Using this information and dimensional reasoningonly, find, to three significant figures, the onlypossible dimensionally homogeneous formula σ=y f ( M , I ).
The spin rate of a tennis ball determines the aerodynamic forces acting on it. In turn, the spin rate is a§ectedby the aerodynamic torque. If the torque depends on áight speed V , density , viscosity , ball diameter D,angular velocity !, and the fuzz height, hf , Önd the important dimensionless variables for this case. Use V ,, and D as your scaling (repeating) variables.
Ius A fluid flow situation depends on the Nelocity (V), the density several lineor dimension, 4shi, h2.pressure drep (DO > gravity (9) , Viscosity , Susface tension ). and bulk mo dulus of elasticity k. Apply dimen sional analysis. to these variables td A s*
A2) In order to solve the dimensional analysis problem involving shallow water waves as in Figure 2, Buckingham Pi Theorem has been used. h Figure 2 Through the observation that has been done, the wave speed © of waves on the surface of a liquid is a function of the depth (h), gravitational acceleration (g), fluid density (p), and fluid viscosity (µ). By using this Buckingham Pi Theorem: a) Analyze the above problem and show that the Froude Number (Fr) and Reynolds Number (Re) are the relevant dimensionless parameters involve in this problem. b) Manipulate your Pi (1) products to get the parameter into the following form: pch := f(Re) where Re = Fr = c) If one additional primary variable parameter involve in this proolem such as, temperature (T). Discuss on the Pi (m) products that can be produce and explain why this dimensional analysis is very important in the experimental work.
When fluid in a pipe is accelerated linearly from rest, it begins as laminar flow andthen undergoes transition to turbulence at a time ttr which depends upon the pipe diameter D,fluid acceleration a, density ρ, and viscosity µ. Arrange this into a dimensionless relationbetween ttr and D.