Chapter 11 DIMENSIONAL ANALYSIS

2 Fundamental theory preliminary expt. Dimensional analysis: ( 因次分析 ) Experiments Practical Problems analytical soln. (e.g.,Eq.(8.7)) numerical soln. data,figures, charts Buckingham pi theorem pertinent dimensionless groups Expt

DIMENSIONS ► Length, L. ► Time, t. ► Mass, M. ► Temperature, T. ► Force, N. (or ) ► Energy, J. (or ) In fluid mechanics, the three (or four) basic dimensions are usually taken to be mass M, length L, and time t and temperature T. In fluid mechanics, the three (or four) basic dimensions are usually taken to be mass M, length L, and time t and temperature T. The dimensional analysis of energy problems will require the addition of two more fundamental dimensions, heat and temperature. The dimensional analysis of energy problems will require the addition of two more fundamental dimensions, heat and temperature. T 11.1 ◎ Fundamental dimensions ： ◎ Deduced dimensions (units)

4 ◎ Dimensional analysis ： to group the variables (in a given situation) into dimensionless parameters (e.g., ). ► Dimensional analysis is a method for reducing the number and complexity of experimental variables which affect a given physical phenomenon CONCEPT OF DIMENSIONAL ANALYSIS

5 Dimensional analysis has several side benefits. ● First ： an enormous saving in time and money. Suppose one knew that the force F on a particular body immersed in a stream of fluid depended only on the body length L, the stream velocity V, the fluid density, and the fluid viscosity ； that is, F 12.5F 7.14F 5.6F 7.17 (5.1)

6 Suppose further that the geometry and flow conditions are so complicated that our integral theories and differential equations fail to yield the solution for the force. Then we must find the function experimentally. Generally speaking, for each variable, it takes about 10 experimental points to define a curve. To find the effect of body length in Eq.(5.1) we shall have to run the experiment for 10 lengths L. For each L we shall need 10 values of V, 10 values of,10 values of, making a grand total of, or 10,000 experiments.

7 At $5 per experiment － well, you see what we are getting into. However, with dimensional analysis, we can immediately reduce Eq.(5.1) to the equivalent form that is, the dimensionless force coefficient (the Euler Number, Eu) is a function only of the dimensionless Reynolds number. We can establish g by running the experiment for only 10 values of the single variable called the Reynolds number. We do not have to vary p or separately but only the Re ( ). F 12.2 or,V,,V,L 

8 This we do merely by varying velocity V in say, a wind tunnel or drop test or water channel, and there is no need to build 10 different bodies or find 100 different fluids with 10 densities and 10 viscosities. The cost is now about $50, maybe less. This we do merely by varying velocity V in say, a wind tunnel or drop test or water channel, and there is no need to build 10 different bodies or find 100 different fluids with 10 densities and 10 viscosities. The cost is now about $50, maybe less. Thus by combining the variables into a smaller number of dimensionless parameters, the work of experimental data reduction and cost are considerably reduced. Thus by combining the variables into a smaller number of dimensionless parameters, the work of experimental data reduction and cost are considerably reduced. F 7.21

9 ˙ A second side benefit of dimensional analysis is that it helps our thinking and planning for an experiment or theory. Dimensional analysis will often give a great deal of insight into the form of the physical relationship we are trying to study, i.e., easier to interpret experimental data and to understand the phenomena. Dimensional analysis will often give a great deal of insight into the form of the physical relationship we are trying to study, i.e., easier to interpret experimental data and to understand the phenomena. #51

10 ˙ A third benefit is that dimensional analysis provides scaling laws which can convert data from a cheap, small model into design information for an expensive, large prototype. Example ： airplane lift force similarity million-dollar airplane a small model F 10.2

11 Certain aspects of similarity will be used to predict the flow behavior of equipment on the basis of experiments with scale models. In the simple case of Eq.(5.1), similarity is achieved if the Reynolds number is the same for the model and prototype because the function g then requires the force coefficient to be the same also ： where m and p mean model and prototype, respectively. From the definition of force coefficient, Equation (5.4) is a scaling law ： if you measure the model force at the model Reynolds number, the prototype force at the same Reynolds number equals the model force times the density ratio times the velocity ratio squared times the length ratio squared. If, then (5.3) (5.4)

12 Dimensional Analysis Of The Navyer-stokes Equation For a differential equation describing a given flow situation ► Dimensional homogeneity ： each term in the equation has the same units. ► The ratio of one term to another must be dimensionless. ► We are then able to give some physical interpretation to the dimensionless parameters thus formed DIMENSIONLESS PARAMETERS ： FORMATION AND PHYSICAL INTERPRETATION ◎ Formation of dimensionless group ► experiment ： The Buckingham method ► theory ： The momentum equation

13 ► A classic example ： the Navier-Stokes equation ► The physical meaning and expression of each term are as follows ： each of these has dimensions of (9-19) Acceleration or inertial Force, Gravity force, Pressure force, Viscous force,

14 ► Dividing each of the terms on the right-side of equation (9-19) by the inertia forces, we form the following dimensionless parameters ： pressure force inertial force viscous force inertial force gravity force inertial force (= )

15 ► They or their reciprocals appear often in fluid analysis, and are given special names as follows ► They or their reciprocals appear often in fluid analysis, and are given special names as follows inertiaforce inertia force gravity force = Fr, the Froude number (11-1) pressure force inertia force = Eu, the Euler number (11-2) viscous force inertia force = Re, the Reynolds number (11-3) dimension -less

16 ► The dimensional variables making up these parameters will vary with the particular situation. For instance, the significant length might be the diameter of a cylinder or the distance from the leading edge of a flat plate. ► To avoid confusion it is advisable to specify clearly the reference length, reference velocity, and so on when reporting value for any dimensionless parameter. Two systems are said to be dynamically similar if the pertinent dimensionless numbers are equal. Two systems are said to be dynamically similar if the pertinent dimensionless numbers are equal. Dynamic similarity is a fundamental requirement in extending experimental data from a model to its prototype. Dynamic similarity is a fundamental requirement in extending experimental data from a model to its prototype. F 9.7F 9.6 ModelPrototype Scale up or down dimensionless numbers are equal

17 The Buckingham Method ► When there is no governing differential equation, a more general procedure is required ： Buckingham method. (1) listing of the variables significant to a given problem. (2) determine the number of dimensionless parameters into which the variables may be combined ： Buckingham pi theorem. ► Buckingham pi theorem ˙ The number of dimensionless groups used to describe a situation involving n variables is equal to n － r, where r is the rank of the dimensional matrix of the variables METHOD OF DIMENSIONAL ANALYSIS ：

18 Dimensional matrix ： the matrix formed by tabulating the exponents of the fundamental dimensions M, L, and t, which appear in each of variables involved. Dimensional matrix ： the matrix formed by tabulating the exponents of the fundamental dimensions M, L, and t, which appear in each of variables involved. i=n － r (11-4) i = the number of independent dimensionless groups n = the number of variables involved r = the rank of the dimensional matrix

19 F = f (L, V, ρ, μ) dimensional analysis How many dimensionless group ？ What are they ？ Eu = g (Re) experiment CfCf Re F 7.14F 7.17 F 12.2 #50 Eu ： Euler number Eu ≡ C f (force coefficient) Re ： Reynolds number ReEu

20

21

22 Π group

23

24

25

26 ► This example problem shows that dimensional analysis has enabled us to relate the original five variables in terms of only two dimensionless parameters in the form. F 12.4F 12.2

MODEL THEORY ► In the design and testing of large equipment involving fluid flow it is customary to build small models geometrically similar to the larger prototypes. Experimental data achieved for the models are then scaled up to the full-sized prototypes according to the requirements of geometric, kinematic, and dynamic similarity. ModelPrototype geometric kinematic dynamicsimilaritysimilarity similarity An application of experimental data achieved for a model to a full-sized prototype require that certain similarities exist between the model and prototype. ModelPrototype similarity

28 ► Geometric similarity ： the ratio of significant dimensions is the same for the two systems. For example, in Figure (For complicated geometries, geometric similarity would, require all geometric ratios to be the same between a model and prototype.) (For complicated geometries, geometric similarity would, require all geometric ratios to be the same between a model and prototype.) F 11.1 Kinematic similarity in geometrically similar systems (1) and (2), the velocities at the same points are related according to the relations a requirement of kinematic similarity is the geometric similarity also exists. Kinematic similarity in geometrically similar systems (1) and (2), the velocities at the same points are related according to the relations a requirement of kinematic similarity is the geometric similarity also exists. ► Kinematic similarity

29 ► Dynamic similarity ˙Two systems are said to be dynamically similar if the pertinent dimensionless numbers are equal. ˙Dynamic similarity is a fundamental requirement in extending experimental data from a model to its prototype. ModelPrototype Scale up or down dimensionless numbers are equal When scaling ( scale-up or scale-down ), all pertinent dimensionless parameters must be equal between the model and prototype. In working on a problem, however, use of equality of only one of them (e.g., Re m =Re p ) is often sufficient to obtain the answer.

30 ※ Drag force in high-speed gas flow Eu = f (Re, M) or C F = f (Re, M) M (Mach number) ≡ gas velocity / sound velocity

31 wind tunnel

32 wind tunnel

33

34

35

36

37

38

39

40

41

42

43 Model in nitrogen at 5 atm, 183K (wind tunnel) Prototype in air at sea level scale up similarity (= 3.26 m) L m = ？ (= 3.26 m) (= 4.85m/s) V m = ？ (= 4.85m/s) V sound,m = 275m/s μ = 1.2×10 -5 Pa.S (7-10) geometric similarity Kinematic similarity dynamic similarity L p = 24.38m V p = 60m/s M m =M p Re m =Re p Eu m =Eu p ρ = kg/m 3 V sound,m =340m/s μ = 1.789×10 -5 Pa.S ρ = kg/m 3 F=f (v, ρ, μ, L) analysis dimensional E μ =g (Re) E μ =g (Re, M) high speed

44 F, L, V, ρ, μ Core group: (1)number of variables (2)contains all fundamental dimensions (3) exclude F Π 1 = V a ρ b L c Π 2 = V d ρ e L f i = n-r number of variables ： n=5 number of fundamental dimensions ： r=3 number of dimensionless groups: i = 5-3 = 2 number of dimensionless groups i=2

45 F A F L2L2 ρ U ∞ 2 2 = C f ρv 2 = = (12-2) CfCf

Table_11-2 Dimensionless time scale, t* tυ L distance of traveling characteristic length

(laminar or turbulent) Flow regularity = Inertial force viscous force laminar flow turbulent flow ρ*3 D*3 V*3 or/andμ*1/3 ρ↓,D↓,V↓ or/and μ↑ ρ↑,D↑,V↑ or/and μ↓