Fluid Mechanics Fundamentals EBS 103 /HYD 103N. Winter quarter, 4 units

Lecture TWRF 11:00-11:50, Giedt 1006; Office hours T-F 12-1; 221 Veihmeyer Hall.

 

Course Description: Axioms of fluid mechanics, fluid statics, kinematics, velocity fields for one-dimensional incompressible flow including boundary layers, turbulent flow time averaging, dimensional analysis, and macroscopic balances to solve a range of practical problems.

Concepts: Continuum approach to deforming physical/biological systems, transport theorem integral analysis, stress vector and stress tensor analysis, microscopic and macroscopic analysis of mass as well as linear and angular momentum, downscaling for information retrieval, application of theory to solve practical problems.

Goal: To apply knowledge of mathematics, science and engineering to natural and engineered systems.  To use engineering methods to identify, formulate and solve problems.  Prepare for study of heat and mass transfer in physical/biological systems.

Prerequisites: PHY 9B and MTH 21D (MTH 22A and 22B recommended).

Instructor(s):

Wes Wallender, Professor, 221 Veihmeyer Hall, wwwallender@ucdavis.edu, 752.0688.

       Kelly Williams, Reader, 3022 Bainer Hall, kcwilli@ucdavis.edu, Office hours 2-4 pm Mondays.

Text: Introduction to Fluid Mechanics. S. Whitaker. R.E. Kreiger Publishing Co. 1982.

Grading: Two midterms 20% each, final exam 40%, homework 20% (Assigned homeworks for week are due the following Tuesday, no credit if late).

 

Brief Course Outline:

Axioms of Fluid Mechanics

Mass and Momentum Principles, Vector Invariance, Stress Vector, Stress Tensor.

Statics

Fluids at Rest, Forces on Submerged Surfaces.

Kinematics

Transport Theorems and Mass Conservation, Application of Macroscopic Mass Balance, Cauchy’s First and Second Equations, Viscous Stress, Rate of Strain and Vorticity Tensors, Physical Interpretation of the Rate of Strain and the Vorticity Tensors, Newton’s Law of Viscosity, the Equations of Motion, Navier Stokes Equation, Applications.

Empiricism

Dimensional Analysis, Transition and Turbulent Flow, Time Averaged Continuity and Navier Stokes Equation, Physical Interpretation of Turbulence, Eddy Viscosity and Prandtl’s Mixing Length Theory, Application to Turbulent Pipe Flow.

Macroscopic Balances and Downscaling

Bernoulli’s Equation, Moving Control Volumes and Inertial Frames, Mechanical Energy Balance, Applications, Turbulent Flow in Pipes, Friction Factors, Pipeline Design.

 

 

Prepared by Wes Wallender, December, 2009

Expanded Course Outline, Winter 2009.

 

Date

Lecture/Topic

Study

Homework

01.05

1. Mass and momentum principles.  A continuous material body and the Euler Cut are introduced and the stress vector is shown to be a direction dependent vector.  Mass, linear momentum and angular momentum conservation principles are reviewed and Euler’s laws are given as axioms.  Relations between Newton’s laws to Euler’s laws are developed.

Sec. 1.3, pages 32-36 and Ch. 1 in Fundamentals of Fluid Mechanics by Whitaker, 1982.

 

Sec. 4.3 and Ch. 13 in Calculus and Analytic Geometry by Stein and Barcellos, 1992 or Sec. 3.3 and Ch. 13 in Thomas’ Calculus by Weir, Hass and Giordano, 2006.

HW 1.

Apply Euler’s Second Law to a two body system using Euler cuts around body I, body II and bodies I and II. 

Show that Euler’s second law is restricted to the strong form of Newton’s third law (central force). Assume the separated bodies are small such that .  For example

01.06

2. Vector invariance.  Reference and inertial frames are defined.  Transformations of base vectors and components of vectors arise from invariance.  The summation and free index notation as well as the Kronecker delta are powerful tools used in mechanics.

Sec. 1.6.

HW 2.

1. Using vector invariance, show how primed basis vectors can be transformed into unprimed basis vectors.

2. Show how to find the transformation of vector components from the primed to the unprimed coordinate system.  (Hint: use orthogonality condition shown in class).

3. Using vector invariance, show how unprimed basis vectors can be transformed into primed basis vectors.  Find another orthogonality condition starting with the primed basis vectors.

01.07

3. Projection, vectors and tensors.   The projected area theorem as well as the vector projection operator (tensor), with which you are already familiar, are tools necessary to understand the development of Cauchy’s Fundamental Theorem. 

Ch. 12 in Calculus and Analytic Geometry or Ch. 12 in Thomas’ Calculus.

HW 3.

Show that a•B = BT•a, in which a is a first order tensor (a vector) and B is a second order tensor, by expanding the left hand side using mixed notation.

01.08

4. Stress vector and tensor.    Stress is a doubly directed quantity and Cauchy’s lemma reveals its nature and supports the development of Cauchy’s Fundamental Theorem which provides the relationship between the stress tensor and the stress vector. 

Sec. 4.2

 

01.12

5. Static Fluid.  Fluid under no shear stress is static and the normal to the surface and the stress vector are co-linear.  The gradient, divergence and Stokes theorems are the relations used to find the point or field equations of mass and momentum conservation from the integral equations.  These equations are integrated for arbitrary control volumes to provide the density, pressure and velocity fields.

Sec. 2.2 and 3.3.

HW 4.

1. On the left hand side substitute mixed notation, apply the gradient theorem and convert back to Gibbs notation to show the right hand side.

01.13

6. Forces on Submerged Surfaces.  Euler’s first equation is integrated to calculate the forces on plane and curved surfaces, and to derive Archemides Principle.  The projected area theorem simplifies the calculations for complex geometries.

Sec. 2.3-2.7.

HW 5.

Problems 2.1 and 2.8

01.14

7. Kinematics.  Material and spatial descriptions of moving particles are the foundation for determining their position, velocity and acceleration during deformation.  When the identified particles in the observed system do not change, the derivative is defined as the material derivative.

Pages 75-84.

HW 6.

Problem 3.2

01.15

8. Kinematics. Streamlines, path lines and streak lines differ according to the particles observed and tracked.  If derivatives are taken with respect to time for an arbitrarily moving observer, the derivative is defined as the general derivative, not material derivative.

Pages 97-98.

HW 7.

Problem 3.1

01.19

9. Kinematics. Physical interpretation of the rate of strain and the vorticity tensors.  From the velocity gradient calculate the rate of stretching of a line element, the rate of angle change between material line elements and the rate of rotation (rigid body rotation).  Relate the vorticity tensor and vorticity vector.

Pages 135-138 and Sec. 5.3.

HW 8.

For the plane Couette flow illustrated in Figure 5.3-1, find the rate of strain in the θ direction and find θ that maximizes rate of strain which is the rate of decrease of the angle between unit vectors initially in the x and y directions.

01.20

10. Kinematics. Physical interpretation of the vorticity tensors.  From the velocity gradient calculate the rate of rotation (rigid body rotation).  Relate the vorticity tensor and vorticity vector.

 

HW 9.

For the plane Couette flow illustrated in Figure 5.3-1, find the components of the vorticity vector.

01.21

11. Transport Theorems and Mass Conservation.  Time derivatives of material volume integrals have already appeared in Euler’s first and second laws as well as the mass conservation equation.  To develop the microscopic point equations of motion and mass conservation, the order of differentiation and differentiation must be reversed, and in so doing time derivatives of volume integrals can be formed as integrals of derivatives.  Furthermore, in macroscopic analysis presented later in the course, derivatives of volume integrals which are not material, will appear and the General Transport Theorem (GTT) will be used to reverse the operations.  Leibniz Rule, also used later in the course, is the GTT in one dimension.

Secs. 3.4 and 3.5.

HW 10.

Show the development the Special form of the Reynold’s Transport Theorem (Hint: let ).

01.22

12. Application of Macroscopic Mass Balance.  Running in the rain.

Sec. 7.1.

HW 11.

Flow in veins and arteries is a transient process in which elastic conduits expand and contract.  Consider an artery having a radial velocity at the inner radius of 0.012 cm/s.  The length L of the artery is 13 cm and the volumetric flow rate at the entrance of the artery is 0.3 cm3/s.  At some instant in time, the inner radius of the artery is 0.15 cm.  At that particular moment, what is the volumetric flow rate at the artery exit? (hint:  dA=rdθL in which A is cross sectional area, r is the radius and θ is the angle in radians.)

01.26

13. Cauchy’s First and Second Equations.  Representing the stress vector in Euler’s first law as a function of the stress tensor allowed the transformation of the area integral into a volume integral.  The special form of the Reynolds transport theorem applied to Euler’s first law transforms the integral equation into the point or field equation of linear momentum including the stress tensor. The resulting point linear momentum equation is Cauchy’s first equation.  The divergence theorem, Reynolds transport theorem, Cauchy’s lemma and Cauchy’s first equation transform Euler’s second law, an integral equation, into the point or field equation showing the symmetry of the stress tensor.  This is Cauchy’s second equation.

 

HW 12.

Use the special form of the Reynolds transport theorem to show that

01.27

14. Newtonian Fluid, Newton’s Law of Viscosity, Viscous Stress, Rate of Strain and Vorticity Tensors.  The viscous stress tensor is defined as a function of pressure and the rate of deformation of the fluid via the rate of strain tensor.  The velocity gradient tensor is decomposed into the symmetric rate of strain tensor and the vorticity tensor which does not contribute to the rate of deformation of the fluid.

Pages 14-16, 133.

 

01.28

15. Newton’s Law of Viscosity and the Equations of Motion.  The linear tensor equation of viscosity for Newtonian fluids is substituted into the stress equations of motions.  These simplify to the Navier-Stokes Equation if flow is incompressible.

Pages 128-134 and Sec. 5.4.

 

01.29

MIDTERM (Through Lecture 15)

 

 

02.02

16. Application.  Uniformly accelerated unconfined flow.  Analytical solutions are available for confined and unconfined flows.  In general both velocity and pressure are dependent variables in each of the motion equations.  For relatively simple flow problems presented in Lectures 16 through 20, separate pressure and velocity differential equations lead to exact and approximate analytic solutions. 

Pages 166-169.

HW 13.

Problem 5-12.

02.03

17. Application.  One-dimensional steady laminar flow confined in a tube.

Pages 169-173.

HW 14.

Problem 5-14.

02.04

18. Application.  Steady laminar boundary layer flow.  Order of Magnitude Estimates of terms in the conservation equations are developed in preparation for simplifying the mass conservation and motion equations to the Prandtl Boundary Layer (PBL) equations for laminar flow.  A process of Intuit, Estimate, Restrict, Constrain Assume (IERCA) is introduced.

Sec. 11.2.

HW 15.

Problem 11-1.

02.05

19. Application.  Steady laminar boundary layer flow.  Via the IERCA process we discover that the length Reynolds number must greatly exceed unity to apply the PBL equations.

Class notes.

 

02.09

20. Application.  The von Karman-Pohlhausen integral method provides an approximation of the velocity profile for laminar boundary layer flow.  In contrast to the exact solutions we developed for the velocity and pressure fields by separation of variables, this is an approximate solution.  The solution is approximate because terms in the original motion equations are eliminated, because the boundary layer thickness is approximate (not a material surface) and the velocity profile function is pre-determined.

Class notes.

HW 16.

Starting with the integral motion equation

express the velocity component as a third order polynomial in  and show that .

02.10

21. Dimensional Analysis.  For more complex flows for which only numerical solutions may or may not be available, model experiments are useful.  The governing equations are made dimensionless to reduce the number of experiments needed to solve complex flow problems that are not susceptible to analysis.  Dimensionless velocity and pressure depend on the Reynolds number but they are independent of the Froude number for confined flows.

Sec. 5.5.

HW 17.

Work the racing sloop example problem which starts on page 163.  Show all the steps, do not just copy what is in the text.  Discuss geometric and dynamic similarity.

02.11

22. Transition and Turbulent Flow.  In the laminar boundary layer fluid parcels follow a straight path, deform and rotate while in the transition region the path is curvilinear and parcels oscillate.  In the turbulent region, the path is undefined and the parcel rotates unpredictably.  For confined and unconfined flows small disturbances in the laminar flow region create velocity variations in time.  Velocity is decomposed into time average and turbulent fluctuation terms.  Because the time scales for each term are disparate, the time average of the time average velocity is equal to the time average velocity.

Sec. 6.1 and class notes.

HW 18.

Problem 6.1

02.12

23. Time Averaged Continuity and Navier Stokes Equations.  Leibniz rule is used, once again, to derive the time averaged equations of incompressible flow.

Sec. 6.2.

HW 19.

Problem 6.2

02.16

24. Time Averaged Continuity and Navier Stokes Equations.  The time-averaged equations of motion indicates that turbulent flow can be treated in the same way as laminar flows provided the pressure and velocity are replaced by the time-averaged quantities and the viscous stress tensor is replaced by the total time-averaged stress tensor which is the sum of the viscous and turbulent stress tensors.

Sec. 6.2

 

02.17

25. Physical Interpretation of Turbulence.  Turbulence is generated near the tube wall and the intensity falls off toward the center of the tube.  In the central region the generating force (shear deformation) decreases and viscous forces tend to reduce turbulence.

Sec. 6.3.

 

02.18

26. Eddy Viscosity and Prandtl’s Mixing Length Theory.  As an analog to laminar flow, a turbulent or Eddy viscosity was developed by Prandtl through a simplified interpretation of turbulent momentum transfer.  Convective and molecular momentum transport are explored.

Sec. 6.4.

 

02.19

27. Application to Turbulent Pipe Flow Velocity Profiles.  Mixing length theory is applied to turbulent pipe flow to calculate the time averaged velocity profile.

Sec. 6.5.

 

02.23

28. Macroscopic Momentum Balance.  To solve more complex problems that are not subject to microscopic analysis, we supplement information from intuition and experimentation, and find solutions which are correct on the average.  The governing equations are satisfied for a control volume rather than point-wise.  Information lost through integration must be replaced by intuition, experiment or analysis at smaller length scales.

Secs. 7.1 and 7.2.

HW 20.

Find Euler’s First Law

starting from the following axiomatic statement of the linear momentum principle

02.24

29. Application.  Jets and Plates.  Force exerted by the fluid on a plate is calculated using the macroscopic momentum conservation principle.  Intuition, order of magnitude estimates and restrictions lead to simplifications of the equations and the corresponding constraint such that inertial terms dominate viscous terms (IEARC).

Secs. 7.1 and 7.8.

HW 21.

Problem 7-6.  Ignore the comments about the energy equation and use the momentum balance to solve this problem.  Microscopic scale information lost by integration over area is recovered or justified by the statement that “viscous surface forces can be neglected.”

02.25

30. Bernoulli’s Equation.  Bernoulli’s is obtained by first extracting the component of the Navier-Stokes equation tangent to a streamline, simplifying that result by neglecting the local acceleration and viscous effects, and then integrating along a streamline.

Pages 230-235.

HW 22.

Problem 7-5 and find the velocity at location 1 in Figure 7.8-3 using Bernoulli’s equation.

02.26

31. Mechanical Energy Balance.  By forming the scalar product with the velocity vector the necessity of evaluating terms in the macroscopic momentum balance equation at solid surfaces is eliminated but a viscous dissipation term arises which must be evaluated experimentally. 

Sec. 7.3.

 

03.02

32. Mechanical Energy Balance.  Keep in mind that the momentum and mechanical energy balances come from the same physical principle but the assumptions in making approximate solutions are different.

Sec. 7.3.

HW 23.

Problem 7-3 using macroscopic mechanical energy balance equation only.

03.03

33. Application.  Sudden expansion in a pipeline.

Sec. 7.5 and pages 311-316.

HW 24.

Problem 7-9.

03.04

34. Turbulent flow in pipes.  We begin developing a consistent method of interpreting experimental data.

Pages 285-293.

 

03.05

Review

 

 

03.09

MIDTERM (Through lecture 32)

 

 

03.10

35. Friction Factors.  Experimental data is interpreted to generalize application of the momentum balance equation to non-circular conduits as well as flows around spheres and cylinders.

Sec. 8.2.

HW 25.

Problem 8-4.

03.11

36. Pipeline Design.  The macroscopic mechanical energy balance equation simplifies calculation of headloss in a pipeline.  The energy and momentum balance equations are combined to arrive at the simplified approach.

Sec. 8.3.

HW 26.

Problem 8-9.

03.12

Review

 

 

03.16

 

FINAL EXAM (Comprehensive)

08:00 am -10:00 am, Giedt 1006