1. 1 MECH 221 FLUID MECHANICS (Fall 06/07) Chapter 6: DIMENTIONAL ANALYSIS Instructor: Professor C. T. HSU

2. MECH 221 – Chapter 6 2 6 Dimensional Analysis  The objective of dimensional analysis is to obtain the key non-dimensional parameters that govern the physical phenomena of flows.  Variables like etc. are combined to form the key parameters that have no physical units (dimensionless). The non-dimensional parameters include both the geometric and dynamic parameters.

3. MECH 221 – Chapter 6 3 6.1 Similarity  Geometric Similarity  Dynamic Similarity  Two flows are said to be similar if they have the same geometric and dynamic dimensionless parameters.

4. MECH 221 – Chapter 6 4 6.1.1 Geometrical Similarity  Two body are geometrically similar, if the geometry of one can be obtained from another by scaling all dimensions by the same factor. B1B1 A1A1 A A2A2 B B2B2 C2C2 C C1C1

5. MECH 221 – Chapter 6 5 6.1.2 Dynamic Similarity  Flows are said to be dynamically similar if by scaling the dependent and independent variables they yield the same non-dimensional parameter.  This is more difficult to achieve than the geometric similarity. What are these non-dimensional parameters? How can they be found?

6. MECH 221 – Chapter 6 6 6.2 Buckingham Pi Theorem  Dimensionless product is the product of several dimensional quantities that render the product dimensionless  The rank of matrix N m is the minimum dimensions that leads to non-zero determinant, which is also the minimum dimensions of the quantities that describe the physics.

7. MECH 221 – Chapter 6 7 6.2 Buckingham Pi Theorem  Given the quantities that are required to describe a physical law, the number of dimensionless product (the “Pi’s”, N p ) that can be formed to describe the physics equals the number of quantities (N v ) minus the rank of the quantities, i.e., N p =N v –N m,

8. MECH 221 – Chapter 6 8 6.2.1 Example  Viscous drag on an infinitely long circular cylinder in a steady uniform flow at free stream of an incompressible fluid.  Geometrical similarity is automatically satisfied since the diameter (R) is the only length scale involved. D D: drag (force/unit length)

9. MECH 221 – Chapter 6 9 6.2.1 Example  Dynamics similarity

10. MECH 221 – Chapter 6 10 6.2.1 Example  Both sides must have the same dimensions!

11. MECH 221 – Chapter 6 11 6.2.1 Example where is the kinematic viscosity.  The non-dimensional parameters are:

12. MECH 221 – Chapter 6 12 6.2.1 Example  Therefore, the functional relationship must be of the form:  The number of dimensionless groups is N p =2

13. MECH 221 – Chapter 6 13 6.2.1 Example  The matrix of the exponents is  The rank of the matrix (N m ) is the order of the largest non-zero determinant formed from the rows and columns of a matrix, i.e. N m =3.

14. MECH 221 – Chapter 6 14 6.2.1 Example  Problems: No clear physics can be based on to know the involved quantities Assumption is not easy to justified.

15. MECH 221 – Chapter 6 15 6.3 Normalization Method  The more physical method for obtaining the relevant parameters that govern the problem is to perform the non-dimensional normalization on the Navier-Stokes equations: where the body force is taken as that due to gravity.

16. MECH 221 – Chapter 6 16 6.3 Normalization Method  As a demonstration of the method, we consider the simple steady flow of incompressible fluids, similar to that shown above for steady flows past a long cylinder.  Then the Navier-Stokes equations reduce to:

17. MECH 221 – Chapter 6 17 6.3 Normalization Method  If the proper scales of the problem are:  Here the flow domain under consideration is assumed such that the scales in x, y and z directions are the same U L P:

18. MECH 221 – Chapter 6 18 6.3 Normalization Method  Using these scales, the variables are normalized to obtain the non-dimensional variables as:  Note that the non-dimensional variable with “*” are of order one, O(1).  The velocity scale U and the length scale L are well defined, but the scale P remains to be determined.

19. MECH 221 – Chapter 6 19 6.3 Normalization Method  The Navier-Stokes equations then become: where is the unit vector in the direction of gravity which is dimensionless.

20. MECH 221 – Chapter 6 20 6.3 Normalization Method  The coefficient of in the continuity equation can be divided to yield  Dividing the momentum equations by in the first term of left hand side gives:

21. MECH 221 – Chapter 6 21 6.3 Normalization Method  Since the quantities with “*” are of O(1), the coefficients appeared in each term on the right hand side measure the ratios of each forces to the inertia force. i.e., where Re is called as Reynolds number and F r as Froude number.

22. MECH 221 – Chapter 6 22 6.3 Normalization Method  The dynamic of fluid motion then depends solely on the magnitudes of these non- dimension parameters, i.e., pressure coefficient and gravitational body-force coefficient.  Flows are dynamically similar if they have the same Re and Fr

23. MECH 221 – Chapter 6 23 6.3 Normalization Method  Remark : New non-dimensional parameters can also emerge from the non-dimensional analysis on the boundary conditions which is not deliberated here. The dimensional analysis reduces experimentalists the need of carrying out measurements for different U, D, etc.

24. MECH 221 – Chapter 6 24 6.4 Characteristics of Non-Dimensional Parameters  Reynolds Number  Froude Number

25. MECH 221 – Chapter 6 25 6.4.1 Reynolds Number  Let’s for simplicity consider the case where the gravitational force has no consequence to the dynamic of the flow, i.e. the case where or the contribution of is only to the static pressure. Then, the pressure P represents the dynamics pressure.  The normalized momentum equation becomes

26. MECH 221 – Chapter 6 26 6.4.1 Reynolds Number where is the viscous diffusion length in an advection time interval of.  Here, measures the time required for fluid travel a distance L.

27. MECH 221 – Chapter 6 27 6.4.1 Reynolds Number  High Reynolds Number Flow, Re>>1  Intermediate Reynolds Number Flow, Re~1  Low Reynolds Number Flow, Re<<1

28. MECH 221 – Chapter 6 28 6.4.1.1 High Reynolds Number Flow  When, inertia force is much greater than viscous force, i.e., the viscous diffusion distance is much less than the length L.  Viscous force is unimportant in the flow region of, but can become very important in the region of near the solid boundary.  This flow region near the solid boundary is called an boundary layer as first illustrated by Prandtl.

29. MECH 221 – Chapter 6 29 6.4.1.1 High Reynolds Number Flow  Flow in the region outside the boundary layer where viscous force is negligible is inviscid. The inviscid flow is also called the potential flow. U Boundary layer flowPotential flow

30. MECH 221 – Chapter 6 30 6.4.1.1 High Reynolds Number Flow  The normalized dimensionless equation to the first order approximation is:  Clearly, the proper pressure scale should be chosen such that is of O(1), and for simplicity, can be set as is the proper pressure scale for high Reynolds number flow.  Flows in the boundary layer are governed by boundary layer equations that need to be derived separately with different approach

31. MECH 221 – Chapter 6 31 6.4.1.1 High Reynolds Number Flow  For inviscid, incompressible, steady flow, the governing equations in terms of dimensional variable to the first order approximation are written as,  If the gravitational force is retrieved, then we have where is the steady Euler equations for incompressible fluid as given in Chapter 2.

32. MECH 221 – Chapter 6 32 6.4.1.2 Intermediate Reynolds Number Flow  When, inertia forces and viscous forces are of equal importance. The flow is viscous in a region of surrounding the body since. No approximation can be done. L U

33. MECH 221 – Chapter 6 33 6.4.1.2 Intermediate Reynolds Number Flow  The governing equations remain as: whose solutions satisfying proper boundary conditions can usually only be obtained numerically.

34. MECH 221 – Chapter 6 34 6.4.1.3 Low Reynolds Number Flow  When, the inertia force is very much smaller than the viscous force. The viscous diffusion length is much larger than L.  The flow is viscous for almost the entire region except at vary far away from the solid boundary.

35. MECH 221 – Chapter 6 35 6.4.1.3 Low Reynolds Number Flow U L

36. MECH 221 – Chapter 6 36 6.4.1.3 Low Reynolds Number Flow  Since implies, the inertia force is negligible and the pressure force has to balance the viscous force. Therefore, the proper scale for P is such that  The governing equation for low Reynolds number flows, without the gravitational force, can be approximated to the first order by

37. MECH 221 – Chapter 6 37 6.4.1.3 Low Reynolds Number Flow  If the gravitational force is recapped, the governing equations to the first order approximation becomes:  Flows of low Reynolds number are called the Stokes flows, or creeping flows.  One such example is the settling of small particles in water. Lava flows from volcanic eruption are also typical low Reynolds number flows although the viscosity is non-Newtonian.

38. MECH 221 – Chapter 6 38 6.4.1.4 Dynamically Similar & Reynolds Number  Flows with the same Reynolds number are dynamically similar.  For example, flight of very small insects in air can be studied much more easily on large models in very viscous fluid (liquids).  Similarly, flows for large object such as train, airplane, tall buildings, etc., can be studied experimentally with small models. (Note: They should be geometrically similar)

39. MECH 221 – Chapter 6 39 6.4.2 Froude Number  The Froude number measures the gravitational effect on the flows. It depends on the problem encountered.  For instance in free surface flows, is the phase speed c of shallow water gravity waves when L is water depth. The Froude number becomes:

40. MECH 221 – Chapter 6 40 6.4.2 Froude Number  If U represents the speed of a ship moving on a flat water surface, the wave patterns generated by the ship, called “ship wakes”, then depend on whether the Froude number is. U c Fr >1 Fr =0Fr <1 Ship wakes

41. MECH 221 – Chapter 6 41 6.4.2 Froude Number  Another example is the rising of a hot-air balloon, which is characterized by the net buoyancy force. The natural convection flows generated by a vertical heated surface represents another example.  g

42. MECH 221 – Chapter 6 42 6.4.2 Froude Number  The ratio of the net buoyancy force (per unit volume) to the inertia force (per unit volume) is: where Ri is the Richardson number. The Richardson number is the key parameter in dealing with advection of fluid of different density.  Another example is the propagation internal waves in stratified fluids.

43. MECH 221 – Chapter 6 43 6.5 Map based on Non-Dimensional Parameter Flow ln Fr ln Re Fr~0 Re~0 Fr~1 Re~1