Citation |

- Permanent Link:
- http://digital.auraria.edu/AA00002581/00001
## Material Information- Title:
- Fluid motion due to a bending wave
- Creator:
- Raskob, Jr., Anthony William
- Publication Date:
- 1998
- Language:
- English
- Physical Description:
- xi, 73 leaves : ; 28 cm
## Subjects- Subjects / Keywords:
- Hydrodynamics ( lcsh )
Fluid mechanics ( lcsh ) Navier-Stokes equations ( lcsh ) Wave-motion, Theory of ( lcsh ) Fluid mechanics ( fast ) Hydrodynamics ( fast ) Navier-Stokes equations ( fast ) Wave-motion, Theory of ( fast ) - Genre:
- bibliography ( marcgt )
theses ( marcgt ) non-fiction ( marcgt )
## Notes- Bibliography:
- Includes bibliographical references (leaves 72-73).
- General Note:
- Department of Mechanical Engineering
- Statement of Responsibility:
- by Anthony William Raskob, Jr.
## Record Information- Source Institution:
- University of Colorado Denver
- Holding Location:
- Auraria Library
- Rights Management:
- All applicable rights reserved by the source institution and holding location.
- Resource Identifier:
- 41471095 ( OCLC )
ocm41471095 - Classification:
- LD1190.E55 1998m .R37 ( lcc )
## Auraria Membership |

Full Text |

Fluid Motion Due to a Bending Wave
by Anthony William Raskob, Jr. B.S., University of Colorado, 1982 A thesis submitted to the University of Colorado at Denver in partial fulfillment of the requirements for the degree of Master of Science Mechanical Engineering 1998 1998 by Anthony William Raskob, Jr. All rights reserved. This thesis for the Master of Science degree by Anthony William Raskob, Jr. has been approved Ken Ortega /cV Date Raskob, Anthony William Jr. (M.S., Mechanical Engineering) Fluid Motion Due to a Bending Wave Thesis directed by Professor Samuel Welch ABSTRACT The fluid motion caused by a bending wave on the boundary of a viscous, incom- pressible fluid is presented and analyzed. The motion is input in the form of a stand- ing wave applied to the boundary of a semi-infinite fluid. A scaling law analysis of the Navier-Stokes and continuity equations is presented which gives the con- ditions under which a linearized form of the Navier-Stokes equations may be ex- pected to yield useful approximate solutions. A linearized solution is obtained and presented. A numerical solution to the full Navier-Stokes equations is obtained us- ing the commercial solver FLUENT running an implementation of the SIMPLE al- gorithm. Comparison of the numerical solution to the linearized solution reveals significant differences in the near-boundary fluid motions. These differences are attributed to the non-linear fluid behaviour not accounted for in the linearized equa- tions. This abstract accurately represents the content of the candidals mend its publication. Signed IV CONTENTS Chapter 1. Introduction 1 1.1 Problem Statement................................................ 1 1.2 Previous Research by Others...................................... 3 1.2.1 Inviscid fluid................................................... 3 1.2.2 Viscous fluid 5 2. Linear Theory Solution 7 2.1 Scaling Analysis 7 2.1.1 Non-dimensionalization using bending wavelength.................. 8 2.1.2 Non-dimensionalization using viscous wavelength................. 11 2.2 Linearized Theory Results 11 2.2.1 Solution to linear momentum equations........................... 12 2.2.2 Fluid behaviour according to linear solution 14 2.3 Consideration of Non-linear Effects............................. 21 v 3. Numerical Solution 23 3.1 SIMPLE Algorithm................................................ 23 3.2 FLUENT Implementation of Bending Wave Problem................. 23 3.2.1 Selection of mesh parameters.................................... 24 3.2.2 Use of symmetry; mesh boundary conditions...................... 26 3.2.3 Moving mesh forcing function 26 3.3 FLUENT Results.................................................. 28 3.3.1 Grid refinement................................................. 28 3.3.2 Time step refinement............................................ 30 3.3.3 Starting transient 30 3.4 Comparison of Linear Theory and FLUENT Results................ 32 3.4.1 Velocity profile results 32 3.4.2 Vector field results 38 4. Conclusions 42 4.1 Directions for further research 42 Appendix A Development of Linearized Solution using Mathematica 43 B. NEWMESH Subroutine Listing 69 vi FIGURES 1.1 Semi-infinite fluid bounded by undulating plate......................... 1 1.2 Sectional view of undulating boundary 2 1.3 Near-field coupling of sub-acoustic plate bending waves 3 1.4 Bending wave coupling to fluid.......................................... 4 2.1 Expression for real portion of u solution to linearized equation set 15 2.2 Pressure contours calculated using linearized theory. 16 2.3 Wall-normal (v) velocity contours calculated using linearized theory. 17 2.4 Wall-normal (u) velocity profile calculated using linear theory (near wall region)..................................................... 17 2.5 Wall-normal (v) velocity profile calculated using linear theory (over- all)........................................................... 18 2.6 Spanwise (u) velocity profile calculated using linear theory (near wall region)..................................................... 19 2.7 Spanwise (u) velocity profile calculated using linear theory (overall). 19 2.8 Spanwise (u) velocity contours calculated using linear theory (near wall region)..................................................... 20 vii 2.9 Spanwise (u) velocity contours calculated using linear theory (over- all)................................................................... 20 2.10 Velocity vector field calculated using linear theory (overall). 21 2.11 Velocity vector field calculated using linear theory (near wall region). 22 3.1 Meshing scheme employed in fluent.............................. 25 3.2 Boundary conditions used with fluent mesh. 26 3.3 Geometry updated after solution of momentum and pressure equa- tions.................................................................. 27 3.4 FLUENT solution time histories for various degrees of mesh reso- lution (u velocity component, x = Ab/2)................................ 29 3.5 FLUENT solution time histories for various degrees of mesh reso- lution (v velocity component, x Ab/4)................................ 29 3.6 FLUENT solution time histories for various degrees of mesh reso- lution (v velocity component, x = Ab/2)................................ 30 3.7 FLUENT solution time histories for various degrees of time step re- finement............................................................... 31 3.8 FLUENT solution time histories for various degrees of time step re- finement (detail)...................................................... 31 3.9 Difference in first and second cycle u velocities (y = 7.2AS). ... 32 viii 3.10 Comparison between first and second cycle u velocities (y = 0.2AS). 33 3.11 Difference in first and second cycle u velocities (y = 0.2AS). 33 3.12 Comparison of linear theory solution and fluent result, u velocity component at 180. 34 3.13 Comparison of linear theory solution and FLUENT result, u velocity component at 315. 34 3.14 Comparison of linear theory solution and FLUENT result, v velocity component at 180. 35 3.15 Comparison of linear theory solution and FLUENT result, v velocity component at 315. 35 3.16 Comparison of linear theory solution and fluent result (dashed), u velocity component near 270, x = Ab/2................ 36 3.17 Comparison of linear theory solution and FLUENT result, near-wall (y = 0.2Ab) u velocity component versus time............ 36 3.18 Comparison of linear theory solution and FLUENT result, u velocity component near 270, x = Xh/4........................... 37 3.19 Comparison of linear theory solution and FLUENT result, v velocity component near 270. x = Ab/2............................. 37 3.20 Schematic illustration of convection-driven near-wall motion. ... 38 IX 3.21 Linear theory solution velocity field at several points through wall motion cycle....................................................... 39 3.22 FLUENT results region showing near-wall effect. 39 3.23 FLUENT calculation near-wall velocity field at several points through wall motion cycle (0-75)......................................... 40 3.24 FLUENT calculation near-wall velocity field at several points through wall motion cycle (90-l 65)...................................... 41 x TABLES 2.1 Typical dimensional values for bending wave experiments........... 10 3.1 Levels of mesh resolution investigated. 25 3.2 Gas law properties of FLUENT. 26 xi Chapter 1 Introduction This work considers the problem of fluid motions induced by wave motion on the surface or boundary of a viscous, incompressible fluid. Specifically, it considers a fluid in contact with a vibrating solid boundary. Because the solution of this prob- lem pertains to fluid motions caused by bending waves on immersed plates, for ease of reference I refer to the boundary vibration as a bending wave. Thus, I consider a semi-infinite fluid, bounded on one side by a boundary which is undulating in a manner similar to that of a vibrating plate (Figure 1.1). Figure 1.1: Semi-infinite fluid bounded by undulating plate 1.1 Problem Statement The bending wave motion takes the form of a single mode wave with charac- teristic wavelength A and radian frequency of vibration u;. In general a sinusoidal wave will be assumed, although other shapes could be considered. Because only a single mode of vibration is considered, and because of the semi-infinite geometry 1 | y deformed shape Figure 1.2: Sectional view of undulating boundary of the problem, coordinate axes can be chosen without loss of generality as shown in Figure 1.1, with the x axis in the plane of the boundary and oriented perpendic- ular to the crests and troughs of the wave pattern, the y axis perpendicular to the plane of the boundary, and the ~ axis in the plane of the boundary and aligned with the waves crests and troughs. A section of the x,y plane at the boundary is shown in Figure 1.2. With this choice of coordinate system, the boundary conditions for the x direction velocity component (it) and y direction velocity component (v) can be expressed: u(.r, y,t)\y=d = 0, i>(.r,t/,i )!>,=,/ = v0g(x) cosut, with d(x,t) = d0g{x)smu>t being the out of plane deflection of the boundary, d0 representing the maximum out of plane deflection, and g(x) is the mode shape function. Note that v0 = d0uj. The first boundary condition arises from the no-slip condition, while in the second is contained the assumption that the boundary is non-porous. I am considering a two dimensional formulation only, therefore iv = 0 for simplicity. Of particular interest is the motion very close to the wall. Experiments have shown that the undulating motion described above can effect significant changes in the structure of the near wall boundary layer. It is in part the purpose of this work to aid in understanding this effect. Since the lengthscales important in viscous flows may span a large range and may include very small lengths, it will be necessary to accurately model the near wall motions. This in turn necessitates accurate consider- ation of these viscous effects which will govern the fluid behaviour in regions close to a solid boundary. 2 1.2 Previous Research by Others This problem touches on several areas of interest, and various aspects of it have been previously considered. Working in the area of sound radiation and control, Cremer [3], Brillioun [1], and others [8, 11] have developed formulations for the far and near field radiation patterns of undulating boundaries. As will be shown in the discussion of Section 1.2.1, these fail, however, to address the effects of fluid viscosity. Likewise, Segal [16] has extensive treatment of fluid surface waves, but employs an inviscid formulation, while Lighthill [7] considers both surface and in- ternal waves, but with limited treatment of viscous effects. Lamb [5] on the other hand does consider internal and surface waves, including the effects of viscosity, as discussed in Section 1.2.2. As will be shown, his treat- ment of free surface waves needs to be modified to account for the solid boundary I am considering. 1.2.1 Inviscid fluid is shorter than the corresponding Figure L3: Near-field coupling of sub- acoustic wavelength, the vibration acoustlc Plate bending waves is sub-acoustic, and creates near- field pressure oscillations in the fluid. These oscillations are partly in a direction perpendicular to the local surface normal of the plate or shell. Thus, for properly oriented bending waves in a shell, spanwise near-field oscillations result. In addi- tion, there is motion of fluid in the wall-normal direction, so that the coupled bend- ing energy produces near-field motions having an elliptical profile (Figure 1.3). For an infinite plate, an expression for the normal velocity due to the existence of a bending wave (Figure 1.4) can be given by (making use of Eulers identity e'e = cos 6 + i sin 6, and use of the real portion of each expression for extracting physical Cremer et al. [3] and others [8, 11] have derived expressions for the pressure field created in a fluid by the action of sub-acoustic vibra- tion modes in a structure in con- tact with the fluid. Vibration in the structure generates vibrations of the same frequency in the fluid, and if, for that frequency, the wave- length of the structural vibration near-field oscillations induced in fluid surface normal (exaggerated) 3 X Figure 1.4: Bending wave coupling to fluid properties is implied): rjy(j) = I>0e tkBl. Cremer [3] gives for the sound pressure radiated into an ambient medium or fluid in contact with the plate an expression of the form: p(.v. y) = p0e~kBTe~lkyy. (1.1) He also gives an expression for the wave equation in the fluid: V2p + k2p = 0. (1.2) Equation 1.1 must satisfy the wave equation, and at the same time, the normal com- ponent of the fluid velocity at the plate surface must be equal to the corresponding normal velocity of the surface (for a non-porous surface with no voids in the fluid). Substituting Equation 1.1 into Equation 1.2 and enforcing the plate-fluid bound- ary condition yields information about the sound pressure above the plate and the wavenumber ky, namely: p(x,y)= , Vol>c e~il(1.3) yj1 *i/*2 with K = k2 kl One can clarify the physical meaning of Equation 1.3 by introducing an angle 6 at which the sound is radiated. This angle must be such that the trace1 of the acoustic wave along the plate has the same wavelength as the plate bending wave [3], or M.e., the projection of the acoustic wave onto the structure 4 sin 6 = A/AB (i.e., sin 9 = k^/k). When AB > A, Equation 1.3 describes a set of plane acoustic waves (Figure 1.4) radiating from the plate. However, when AB < A, there is no real angle at which sound is radiated, and Equation 1.3 instead describes a pressure field which decreases with increasing distance from the plate. In this case, the particle motions can be found to have x and y components: = !kBVo yJk'Q k2 vy = v0e-lkBl'e-'/kl-k2y. This may be interpreted as indicating that the fluid avoids acoustic compression (re- quired for the generation of a sound wave) by moving out of the way laterally, re- sulting in a back and forth sloshing of the fluid [3]. The above represents a potential function solution to the wave equation, and as such, does not enforce the no-slip condition at the wall. Lamb [5] considers a combined potential and stream function solution which more correctly describes the near-wall spanwise motion, as well as accounting for fluid viscosity, but for a certain case of boundary conditions representative of water waves at an air inter- face. 1.2.2 Viscous fluid Lamb begins with the momentum equations linearized in this form: du 1 dP dt dv di p dx 1dP_ p dy + i/V n, g + uV2v in which g is the gravitational constant and v is viscosity. By setting u v p p d(j> dx d
d'tk
suggests the following forms for the potential and stream functions, respectively:
to solve for n such that the auxilliary condition is met: |