index
MANE 6560: Homework
From: Chris Nkinthorn, 2020-01-23
For: Prof. A. Hirsa for Incompressible Flow
Problem 1: Water Jet Stability
Consider the temporal instability of a water jet in the absence of gravity the jet diameter $2R$ is $1 \text{ [mm]}$ in his flowing at $1 \text{ [m/s]}$ based on linear theory
Begin first by linearizing the pressure and velocity fields
$\mathbf{u}=\mathbf{u}_{s}+\mathbf{u}^{\prime}$
$p=p_{s}+p^{\prime}$
These linearized disturbance equations are subsituted into the equations of
Continuity ... $\nabla\cdot{\bf{u}} = 0 $
Momentum ... $\dot{\bf{u}} + \frac{\nabla p'}{\rho} = \alpha T'g \bf{k} +\nu\nabla^2\bf{u} $
the gradient of the static pressure field is identically zero
Use the relationship of energy
Part a: Fundamental Wavelength
use the NS and continuity to find a 6th order ODE describing the viscous momentum balance in terms of position which varies only along the length of the jet, and time
Use the definitions of radii of curvature from calculus
$R_1 = r$
$R{2}=\frac{-\left[1+\left(\frac{\partial r{0}}{\partial x}\right)^{2}\right]^{3 / 2}}{\left(\frac{\partial^{2} r_{0}}{\partial x^{2}}\right)}$
$\left(\frac{\partial}{\partial t}+u \frac{\partial}{\partial x}\right)^{2} r=-\frac{\sigma R}{2 \rho}\left(\frac{1}{R^{2}} \frac{\partial^{2} r}{\partial x^{2}}+\frac{\partial^{4} r}{\partial x^{4}}\right)$
an initial disturbance of the form
$r = a \text{ e}^{kx -\omega t}$ much smaller than the nozzle diameter $10^{-3}R$
Part b: Fundamental Mode
Part c: Volume of Diameter
diameter of a drop is the length of the unstable wavelength found in part a
Part d: Time to pinch off
initial disturbance ... 10E-3 fundamental wavelength
$$
$$
Problem 2: Thermal Convection for Rotating Gap
Rayleigh-Bernard convection in couette flow between rotating cylinders in the narrow gap approximation can be described by similar sets of equations. In the stress free condition, a $6^\text{th}$ order ODE suffices
Taylor Number … $T=\frac{4 d^{4} \Omega_{1} A}{\nu^{2}}$, which relates the rotational centrifugal force to the viscous force in a fluid
Incompresible Flow
Incompresible FlowLecture 1TextbooksIncompressible FlowList of Derivatives Surface Waves and Interfacial PhenomenaIntroduction to waves Continuity EquationLinear dispersive wavesLecture 2Lecture 8: Weakly Nonlinear Waves in Deep Water
Lecture 1
Class 1: 2020-01-13T15:49:925
Covered course outline and such; assigned first homework
Topics Required to do basic research into FM is the typical coursework of
Fluid Mechanics
Viscous Flow & Boundary Layer Theory
Turbulence
Compressible flow
kinetic theory
continuum theory of fluids
combustion aerodynamics
part of the big picture this is one of the topics
Textbooks
Lighthill waves in fluids
Incompressible Flow
Density change is negligible which is seen in low
List of Derivatives
Partial
Material
Total
Surface Waves and Interfacial Phenomena
Introduction to waves
Research in fluid dynamics requires the typical coursework
fluid mechanics
viscous flow and BL theory
turbulence
compressible flow
kinetic theory
continuum mechanics for fluids
speciality courses: aero and combustion
Compressibility so when has M <0.3
for water M less than 0.07 much faster speed of sound for compressibility to act
Continuity Equation
Conservation of mass:
Linear dispersive waves
Described by
k is wave number, related to wave length
euler identity to show exponent is purely imaginary and is the phase function , the amplitude the real component
using dimensionless NS
Lecture 2
Major Equations
euler eqns:
laplacian or harmonic of potential is 0
1d pde
apply to momentum potential and integrate : produces Bernoulli incompressible irrotational flow
at FS for dynamic free surface condition
show dispersion relation: ,
so and for deepwater
shallow water so and . y
Lecture 8: Weakly Nonlinear Waves in Deep Water
due to Rayleigh in a stead motion in coordinate system moving a phase speed c
also
clearly nonlinear
small amplitude but keep high order terms expanded wrt a
substitute into
Ch
Last updated