The drag force FD on a cylinder of diameter d and length l is to be studied. What functional form relates the dimensionless variables if a fluid with velocity V flows normal to the cylinder?
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The drag force FD on a cylinder of diameter d and length l is to be studied. What functional form relates the dimensionless variables if a fluid with velocity V flows normal to the cylinder?
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- Q1: Consider laminar flow over a flat plate. The boundary layer thickness o grows with distance x down the plate and is also a function of free-stream velocity U, fluid viscosity u, and fluid density p. Find the dimensionless parameters for this problem, being sure to rearrange if neessary to agree with the standard dimensionless groups in fluid mechanics. Answer: Q2: The power input P to a centrifugal pump is assumed to be a function of the volume flow Q, impeller diameter D, rotational rate 2, and the density p and viscosity u of the fluid. Rewrite these variables as a dimensionless relationship. Hint: Take 2, p, and D as repeating variables. P e paD? = f( Answer:Problem 4: The power P developed by a wind turbine is a function of diameter D, air density p, wind speed V, and rotational rate @. Viscous effect is negligible. (4a) Rewrite the above relationship in a dimensionless form; (4b) In a wind tunnel, a small model with a diameter of 90cm, rotating at 1200 RPM (revolution per minute), delivered 200 watts when the wind speed is 12m/s. The data are to be used for a prototype of diameter of 50m and wind speed of 8 m/s. For dynamic similarity, what will be (i) the rotational speed of the prototype turbine? (ii) the power delivered by the prototype turbine? Assume air has sea-level density.The time t d to drain a liquid from a hole in the bottom of atank is a function of the hole diameter d , the initial fluidvolume y 0 , the initial liquid depth h 0 , and the density ρ andviscosity μ of the fluid. Rewrite this relation as a dimensionlessfunction, using Ipsen’s method.
- The power P generated by a certain windmill design dependson its diameter D , the air density ρ , the wind velocity V , therotation rate Ω , and the number of blades n . ( a ) Write this relationship in dimensionless form. A model windmill, of diameter50 cm, develops 2.7 kW at sea level when V = 40 m/s andwhen rotating at 4800 r/min. ( b ) What power will be developedby a geometrically and dynamically similar prototype, ofdiameter 5 m, in winds of 12 m/s at 2000 m standard altitude?( c ) What is the appropriate rotation rate of the prototype?The wall shear stress Twin a boundary layer is assumed to be a function of stream velocity U, boundary layer thickness , local turbulence velocity u', density p, and local pressure gradient dp/dx. Using (p, U, and ) as repeating variables, rewrite this relationship as a dimensionless function.A liquid of density ? and viscosity ? is pumped at volume flow rate V· through a pump of diameter D. The blades of the pump rotate at angular velocity ? . The pump supplies a pressure rise ΔP to the liquid. Using dimensional analysis, generate a dimensionless relationship for ΔP as a function of the other parameters in the problem. Identify any established nondimensional parameters that appear in your result. Hint: For consistency (and whenever possible), it is wise to choose a length, a density, and a velocity (or angular velocity) as repeating variables.
- Consider laminar flow over a flat plate. The boundary layer thickness & grows with distance x down the plate and is also a function of free-stream velocity U, fluid viscosity u, and fluid density p. Find the dimensionless parameters for this problem, being sure to rearrange if necessary to agree with the standard dimensionless groups in fluid mechanics.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 DConsider a boundary layer growing along a thin flat plate. The boundary layer thickness & at a downstream distance x is a function of x, the fluid density p, dynamic viscosity, and free stream velocity V. Use Buckingham's theorem with p, x and V as repeating variables, to obtain the relationship between dimensionless parameters Is. Figure 3.2
- A boundary layer is a thin region (usually along a wall) in which viscous forces are significant and within which the flow is rotational. Consider a boundary layer growing along a thin flat plate. The flow is steady. The boundary layer thickness ? at any downstream distance x is a function of x, free-stream velocity V∞, and fluid properties ? (density) and ? (viscosity). Use the method of repeating variables to generate a dimensionless relationship for ? as a function of the other parameters. Show all your work.In the field of air pollution control, one often needs to sample the quality of a moving airstream. In such measurements a sampling probe is aligned with the flow as sketched in Fig. A suction pump draws air through the probe at volume flow rate V· as sketched. For accurate sampling, the air speed through the probe should be the same as that of the airstream (isokinetic sampling). However, if the applied suction is too large, as sketched in Fig, the air speed through the probe is greater than that of the airstream (super iso kinetic sampling). For simplicity consider a two-dimensional case in which the sampling probe height is h = 4.58 mm and its width is W = 39.5 mm. The values of the stream function corresponding to the lower and upper dividing streamlines are ?l = 0.093 m2/s and ?u = 0.150 m2/s, respectively. Calculate the volume flow rate through the probe (in units of m3/s) and the average speed of the air sucked through the probe.1. (a) The motion of a floating vessel through the surrounding fluid results in a drag force D which is thought to depend upon the vessel's speed v, its length I, the density p and dynamic viscosity μ of the fluid and the acceleration due to gravity g. Show that:- D = pv²1² (1) (b) In order to predict the drag on a full scale 50m long ship traveling at 7m/s in sea water at 5°C of density 1027.7225 kg/m³ and viscosity 1.62 x 103 Pa.s, a model 3m long is tested in a liquid of density 805 kg/m³. What speed does the model need to be tested at and what is the required viscosity of the liquid?