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Home , Kinematic similarity, Similitude (model)

ME 305 Mechanics I • Consider that we are interested in determining how the drag acting on a smooth sphere immersed in a uniform flow depends on other fluid and flow variables. Part 7 • Important varibles of the problem are shown below (How did we decide on these?). 푉 Dimensional Analysis and 휇 , 𝜌 퐹퐷 퐷 These presentations are prepared by Dr. Cüneyt Sert Department of Mechanical Engineering • Drag force 퐹퐷 is thought to depend on the following variables. Middle East Technical University 퐹퐷 = 푓(퐷, 푉, 휇, 𝜌) Ankara, Turkey [email protected] • In order to find the actual functional relation we need to perform a set of experiments. You can get the most recent version of this document from Dr. Sert’s web site. • Dimensional analysis helps us to design and perform these experiments in a Please ask for permission before using them to teach. You are NOT allowed to modify them. systematic way. 7-1 7-2

Dimensional Analysis (cont’d) Dimensional Analysis (cont’d)

• The following set of controlled experiments should be done. • It is possible to simplify the dependency of drag force on other variables by using nondimensional (unitless) parameters. • Fix 퐷, 휇 and 𝜌. Change 푉 and measure 퐹퐷. 퐹퐷 𝜌푉퐷 • Fix 푉, 휇 and 𝜌. Change 퐷 and measure 퐹 . = 푓 퐷 𝜌푉2퐷2 1 휇

• Fix 퐷, 푉 and 휇 . Change 𝜌 and measure 퐹퐷. Note : These are only illustrative Nondimensional drag figures. They do not correspond to Nondimensional • Fix 퐷, 푉 and 𝜌. Change 휇 and measure 퐹퐷. any actual experimentation. force (Drag coefficient) (푅푒)

퐹퐷 Constant 퐹퐷 Constant 퐹퐷 Constant 퐹퐷 Constant 퐷, 휇, 𝜌 푉, 휇, 𝜌 퐷, 푉, 휇 퐷, 푉, 𝜌 퐹퐷 𝜌푉2퐷2 Illustrative figure 푉 퐷 𝜌 휇

• We need to perform too many experiments. • Also there are major difficulties such as finding with different , but 푅푒 same . Flow over a sphere at 푅푒 = 15000 7-3 7-4

1 Dimensional Analysis (cont’d) Buckingham Pi Theorem

• To find this new relation, we only need to change the Reynolds number. • Buckingham Pi theorem can be used to determine the nondimensional groups of variables (Pi groups) for a given set of dimensional variables. • We can do it in any way we want, e.g. the simplest way is to change the of air flow in a . • For the flow over a sphere problem studied previously, dimensional parameter set is (퐹 , 퐷, 푉, 휇, 𝜌) and this theorem helps us to find two Pi groups as • All 푅푒 = 15000 flows around a sphere will look like the same and they all provide 퐷 the same nondimensional drag force. It does not matter what fluid we use or how 퐹 𝜌푉퐷 Π = 퐷 and Π = big the sphere is (be aware of very extreme cases). 1 𝜌푉2퐷2 2 휇

• Dimensional analysis is used to formulate a physical phenomenon as a relation • Let’s explain how this works using “the drag force acting on a sphere” problem. between a set of nondimensional (unitless) groups of variables such that the number of these groups is less than the number of dimensional variables. • Step 1 : List all the dimensional variables involved in the problem. • 푛 푛 = 5 • It is important to develop a systematic and meaningful way to perform experiments. is the number of dimensional variables. for our example. • These variables should be independent of each other. For example if the diameter of • Nature of the experiments are simplified and the number of required experiments is a sphere is in the list, frontal area of the sphere can not be included. reduced. • If body are important in a problem, gravitational acceleration should be in the list, although it is a constant.

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Buckingham Pi Theorem (cont’d) Buckingham Pi Theorem (cont’d)

• Step 2 : Express each of the variables in tems of basic dimensions, which are • Step 3: Determine the repeating variables that are allowed to appear in more than one Pi group. 퐿 : length , 푇 : time , 푀 : mass • There should be 푟 many repeating variables. • For problems involving heat transfer Θ () can also be a basic dimension. • If 퐿 is a primary dimension of the problem, we should select one geometric variable • For the example we are studying basic dimensions of variables are as a repeating variable. 푀퐿 퐿 푀 푀 [퐹 ] = , 퐷 = 퐿 , [푉] = , [𝜌] = , [휇] = 퐷 푇2 푇 퐿3 퐿푇 • If 푇 is a primary dimension of the problem, we should select one kinematic variable as a repeating variable. • Our example involves 푟 = 3 primary dimensions. For most problems • If 푀 is a primary dimension of the problem, we should select one dynamic variable 푟 will be 3. as a repeating variable. • Variables having only 퐿 in their dimension are called geometric variables. • Note that this selection is not unique and the resulting Pi groups will depend on our selection. Certain selections are ‘‘better’’ than others. • Variables having only 푇 or both 퐿 and 푇 are called kinematic variables. • For the problem of interest we can select 퐷 , 푉 and 𝜌 as repeating variables. • Variables having 푀 in their dimension are called dynamic variables. • If there is an obvious dependent variable in the problem, do not select it as a • For our example 퐷 is a geometric, 푉 is a kinematic and 퐹 , 휇 and 𝜌 are dynamic 퐷 repeating variable. In our example 퐹 is a dependent variable. We are trying to variables. 퐷 understand how it depends on other variables. 7-7 7-8

2 Buckingham Pi Theorem (cont’d) Buckingham Pi Theorem (cont’d)

• Step 4: Determine (푛 − 푟) many Pi groups by combining repeating variables with • Now determine the Pi group which has 휇 as the nonrepeating variable. nonrepeating variables and using the fact that Pi groups should be nondimensional. 푎 푏 푐 Π2 = 휇 퐷 푉 𝜌 • For our example we need to find 5 − 3 = 2 Pi groups. Each Pi group will include only 푀 퐿 푀 푐 one of the nonrepating variables. • Π should be unitless : − = [퐿]푎 [ ]푏 2 퐿푇 푇 퐿3 푎 푏 푐 Π1 = 퐹퐷 퐷 푉 𝜌 We need to determine Π2 should have no 퐿 dimension : 0 = −1 + 푎 + 푏 − 3푐 푎, 푏 and 푐. 푎 = −1 휇 A nonrepeating Π should have no 푇 dimension : 0 = −1 − 푏 푏 = −1 Π = Unknown combination 2 2 𝜌퐷푉 parameter 푐 = −1 of repeating parameters Π2 should have no 푀 dimension : 0 = 1 + 푐

푀퐿 퐿 푀 푐 • Π should be unitless : − = [퐿]푎 [ ]푏 • Therefore the relation of nondimensional groups that we are after is 1 푇2 푇 퐿3 퐹 휇 Π = 푓 Π → 퐷 = 푓 Π should have no 퐿 dimension : 0 = 1 + 푎 + 푏 − 3푐 1 1 2 𝜌푉2퐷2 1 𝜌푉퐷 1 푎 = −2 퐹퐷 Π should have no 푇 dimension : 0 = −2 − 푏 푏 = −2 Π = 𝜌푉퐷 1 1 𝜌퐷2푉2 • It is better to write the second Pi group as because it is the well known 푐 = −1 휇 Π1 should have no 푀 dimension : 0 = 1 + 푐 Reynolds number. 7-9 7-10

Exercises for Buckingham Pi Theorem Important Nondimensional Numbers of Fluid Mechanics

Exercise : Consider the flow of an incompressible fluid through a long, smooth- • Following nondimensional numbers frequently appear as a Pi group. walled horizontal, circular pipe. We are interested in analyzing the drop, 𝜌푉퐿 푉퐿 • Reynolds number : 푅푒 = = . Ratio of inertia forces to viscous forces. ∆푝, over a pipe length of 퐿. Other variables of the problem are pipe diameter (퐷), 휇 휈 average (푉) and fluid properties (𝜌 and 휇). Determine the Pi groups by a) Δ푝 • Euler number : 퐸푢 = 1 . Ratio of pressure forces to inertia forces. 𝜌푉2 selecting 𝜌 as a repeating parameter, b) selecting 휇 as a repeating parameter. 2 푉 • : 퐹푟 = . Squareroot of the ratio of inertia forces to gravitational Exercise : In a laboratory experiment a tank is drained through an orifice from initial 푔퐿 liquid level ℎ0. The time, 휏, to drain the tank depends on tank diameter, 퐷, orifice forces. diameter, 푑, gravitational acceleration, 푔, liquid properties, 𝜌 and 휇. Determine the 푉 푉 • : 푀푎 = = . Squareroot of the ratio of inertia forces to Pi groups. 퐸푣/𝜌 푐 compressibility forces. Exercise : The diameter, 푑, of the dots made by an ink jet printer depends on the ink 𝜌푉2퐿 𝜌 휇 𝜎 퐷 퐿 • : 푊푒 = . Squareroot of the ratio of inertia forces to surface properties, and , surface tension, , nozzle diameter, , the distance, , of the 𝜎 nozzle from the paper and the ink jet velocity, 푉. Determine the Pi groups. tension forces. 휔퐿 • : 푆푡 = . Used for flows with oscillatory (periodic) behavior. Exercise : The power, 풫, required to drive a propeller is known to depend on the 푉 following variables: freestream speed, 푉, propeller diameter, 퐷, angular speed, 휔, 푝−푝푣 • Cavitation number : 퐶푎 = 1 . Used for possibly cavitating flows. 𝜌 휇 푐 𝜌푉2 fluid properties, and , and the speed of sound . Determine the Pi groups. 2 7-11 7-12

3 Model and Prototype Three Basic Laws of Similitude

• In experimental fluid mechanics we sometimes can not work with real sized objects, • A similitude analysis is done to make sure that the results obtained from an known as prototypes. experiment can correctly be transferred to the real flow field. • Instead we use scaled down (or up) versions of them, called models. • Three basic laws of similitude must be satisfied in order to achieve complete • Also sometimes in experiments we use fluids that are different than actual working similarity between prototype and model flow fields. fluids, e.g. we use regular tap water instead of salty sea water to test the 1. Geometric similarity : Model and prototype must be the same in shape, but performance of a marine propeller. can be different in size. All linear dimensions of the model must be related to corresponding dimensions of the prototype by a constant length ratio, 퐿푟. • It is usually impossible to establish 100 % geometric similarity due to very small details that can not be put into the model. Modeling surface roughness exactly is also impossible.

퐿푝 퐷푝 퐿푟 = = Model Prototype 퐿푚 퐷푚 퐷 www.boeing.com www.reuters.com 푝

Wind tunnel tests of an airliner Race car being tested in a water tunnel 퐷푚 http://www.reuters.com/news/video/story?videoId=131255095 퐿푚 퐿푝 7-13 7-14

Three Basic Laws of Similitude (cont’d) Three Basic Laws of Similitude (cont’d)

2. : Model and prototype flow fields are kinematically similar if 3. Dynamic similarity : Two flow fields should have force distributions such that the at corresponding points are the same in direction and differ only by a identical types of forces are parallel and are related in magnitude by a constant factor constant factor of velocity ratio, 푉푟. of force ratio. • This also means that the streamline patterns of two flow fields should differ by a • If a certain type of force, e.g. compressibility force, is highly dominant in the constant scale factor. prototype flow, it should also be dominant in the model flow. • How suitable would it be to use a water tunnel to study the aerodynamic forces acting on a supersonic missile ?

푉 푉 • If a certain type of force, e.g. surface tension force, is negligibly small in the 푝퐴 푝퐵 푉푟 = = Model prototype flow, it should also be small in the model flow. B 푉푚퐴 푉푚퐵 A B • How suitable would it be to use a very light and very small model to test A Prototype the forces acting on a ship ?

• To establish dynamic similarity we need to determine the important forces of the prototype flow and make sure that the nondimensional numbers related to those forces are the same in prototype and model flows.

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4 Similitude (cont’d) Similitude (cont’d)

Exercise : The aerodynamic drag of a new car is to be predicted at a speed of • The important question is how to decide on the important force types for a given 100 km/h at an air temperature of 25 oC. Engineers build a one fifth scale of the car problem? In other words equality of which nondimensional numbers should be to test in a wind tunnel. It is winter time and the air in the tunnel is 5 oC. Determine sought? how fast the engineers should run the wind tunnel in order to achieve similarity • Reynolds number similarity is important for almost all flows. between the model and the prototype. How each of drag force measured on the model be transferred to the prototype ? • Froude number similarity is important for flows with free surfaces, such as ship resistance, open channel flows and for flows driven by the action of gravity. Prototype Model • Euler number similarity is important mostly for turbomachinery flows with 푉푝 휇푝 , 𝜌푝 considerable pressure changes, for which cavitation may be a concern. 푉푚 휇푚 , 𝜌푚 퐹 푝 퐹푚 • Mach number similarity is important for high speed flows. • Weber number similarity is important for problems involving interfaces with changes and light weighted objects. 퐿푝 퐿푚 • Strouhal number similarity is important for flows with an oscillatory (time periodic) flow pattern, such as Karman vortices shed from bodies. 푅푒푝 = 푅푒푚

𝜌푝푉푝퐷푝 𝜌 푉 퐷 • After deciding on the important parameters of the problem (which is not always = 푚 푚 푚 휇푝 휇푚 an easy task) Buckhingham Pi theorem will also end up with these numbers. 7-17 7-18

Similitude Exercises Similitude Exercises (cont’d)

Exercise : The drag force on a submarine, which is moving well below the free Exercise : The model described in the previous problem will now be used to surface, is to be determined by a test on a model, which is scaled down to one- determine the drag force of a submarine, which is moving on the surface. The twentieth of the prototype. The test is to be carried in a water tunnel. The density properties of the sea water are as given above. The speed of the prototype is 2.6 and kinematic of the seawater are 1010 kg/m3 and 1.3x10-6 m2/s. The m/s. water in the tunnel has a density of 988 kg/m3 and a kinematic viscosity of 0.65x10-6 m2/s. If the speed of the prototype is 2.6 m/s, then determine the a) Determine the speed of the model. a) speed of the model. b) Determine the kinematic viscosity of the liquid that should be used in the experiments. b) ratio of the drag force in the prototype to the one in the model. c) If such a liquid is not available, sea water will be used in the experiments. Neglecting the viscous effects, determine the ratio of the drag force due to the surface waves. This is called incomplete similarity.

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5 Similitude Exercises (cont’d) Similitude Exercises (cont’d) Exercise : A long structural component of a Exercise : A model of a harbor is to be made with a scale ratio of 300. Storm waves bridge has an elliptical cross section. It is having amplitude of 2 m occur on the breakwater of prototype harbor at a speed of known that when an unsteady wind blows 10 m/s. past this type of bluff body, vortices may a) Neglecting the frictional effects, determine the amplitude and speed of the develop on the downwind side that are shed waves in the model. in a regular fashion at some definite frequency. Since these vortices can create harmful periodic forces acting on the structure, it is b) If the tidal period in the prototype is 12 h, determine the tidal period in the important to determine the shedding frequency. For the specific structure of model. interest, 퐷 = 0.1 m, 퐻 = 0.3 m, and a representative wind velocity is 50 km/hr. Standard air can be assumed. The shedding frequency is to be determined through Exercise : An airplane travels in air at a velocity of 200 m/s at an altitude of 5 km. the use of a small scale model that is to be tested in a water tunnel. For the model, o Pressure and temperature of air are 55 kPa and -20 C, respectively. A model of this 퐷푚 = 20 mm and the water temperature is 20 ℃. airplane with a length scale of 10 is tested in a wind tunnel at 20 oC. The specific Determine the model dimension, 퐻 , and the velocity at heat ratio and gas constant for air are 1.4 and 287.1 J/kgK, respectively. Taking the 푚 which the test should be performed. If the shedding effect of compressibility into account, determine the frequency for the model is found to be 49.9 Hz, what is a) velocity of air in the wind tunnel, the corresponding frequency for the prototype? Movie b) density of air in the wind tunnel. Collapse of Tacoma Narrows Bridge http://www.youtube.com/watch?v=j-zczJXSxnw 7-21 7-22

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