AOE 5104 Class 3 9/2/08 Online presentations for today’s class: Vector Algebra and Calculus 1 and 2 Vector Algebra and Calculus Crib Homework 1 due 9/4 Study group assignments have been made and are online. Recitations will be Mondays @ 5:30pm (with Nathan Alexander) Tuesdays @ 5pm (with Chris Rock) Locations TBA Which recitation you attend depends on which study group you belong to and is listed with the study group assignments

Unnumbered slides contain comments that I inserted and are not part of Professor’s Devenport’s original presentation.

Last Class… Vectors, inherent property of direction Algebra Volumetric flow rate through an area Taking components, eqn. of a streamline Triple products, A.BxC, Ax(BxC) Coordinate systems Class 3. 8/29/06 Assignments: Online Presentation “Vector Algebra and Calculus 2” Homework 1, due 9/5/06 Vector Algebra and calculus crib on course web site Study groups

Cylindrical Coordinates Coordinates r,  , z Unit vectors er, e, ez (in directions of increasing coordinates) Position vector R = r er + z ez Vector components F = Fr er+F e+Fz ez Components not constant, even if vector is constant z F ez e er R z  r y x

Spherical Coordinates Errors on this slide in online presentation Spherical Coordinates Coordinates r,  ,  Unit vectors er, e, e (in directions of increasing coordinates) Position vector r = r er Vector components F = Fr er+F e+F e z er e F r e  r  y x

LOW REYNOLDS NUMBER AXISYMMETRIC JET J. KURIMA, N. KASAGI and M. HIRATA (1983) Turbulence and Heat Transfer Laboratory, University of Tokyo

Class Exercise Using cylindrical coordinates (r, , z) Gravity exerts a force per unit mass of 9.8m/s2 on the flow which at (1,0,1) is in the radial direction. Write down the component representation of this force at (1,0,1) b) (1,,1) c) (1,/2,0) d) (0,/2,0) z (9.8,0,0) (-9.8,0,0) (0,-9.8,0) ez e er 9.8m/s2 R z  r y x

Vector Algebra in Components

Fluid particle: Differentially Small Piece of the Fluid Material 3. Vector Calculus Fluid particle: Differentially Small Piece of the Fluid Material

Class 3. 8/29/06 Assignments: Online Presentation “Vector Algebra and Calculus 2” Homework 1, due 9/5/06 Vector Algebra and calculus crib on course web site Study groups

Concept of Differential Change In a Vector. The Vector Field. Scalar field =(r,t) Vector field V=V(r,t) V dV V+dV Differential change in vector Change in direction Change in magnitude

Change in Unit Vectors – Cylindrical System de er+der e ez e+de der P' er e P z er r d 

Change in Unit Vectors – Spherical System  r See “Formulae for Vector Algebra and Calculus” 

Example Fluid particle Differentially small piece of the fluid material The position of fluid particle moving in a flow varies with time. Working in different coordinate systems write down expressions for the position and, by differentiation, the velocity vectors. V=V(t) R=R(t) Cartesian System O Cylindrical System ... This is an example of the calculus of vectors with respect to time.

Vector Calculus w.r.t. Time Since any vector may be decomposed into scalar components, calculus w.r.t. time, only involves scalar calculus of the components

High Speed Flow Past an Axisymmetric Object Finned body of revolution fired from a gun into a supersonic wind tunnel flow for a net Mach number of 2. The plastic shell casing is seen separating. Vincenti, NASA Shadowgraph picture is from “An Album of Fluid Motion” by Van Dyke

Line integrals

Integral Calculus With Respect to Space D(r) D(r) O n B r ds D=D(r),  = (r) dS d Surface S Volume R A Line Integrals For closed loops, e.g. Circulation

For closed loops, e.g. Circulation Mach approximately 2.0 Picture is from “An Album of Fluid Motion” by Van Dyke

Integral Calculus With Respect to Space D(r) D(r) O n B r ds D=D(r),  = (r) dS d Surface S Volume R A Surface Integrals For closed surfaces e.g. Volumetric Flow Rate through surface S Volume Integrals

n dS Mach approximately 2.0 Picture is from “An Album of Fluid Motion” by Van Dyke

Differential Calculus w.r.t. Space Definitions of div, grad and curl In 1-D Elemental volume  with surface S n dS D=D(r), = (r) In 3-D

Alternative to the Integral Definition of Grad We want the generalization of continued

Alternative to the Integral Definition of Grad Cylindrical coordinates continued

Alternative to the Integral Definition of Grad Spherical coordinates

Gradient  = high ndS (medium) ndS (large) n  = low Resulting ndS (small) Elemental volume  with surface S ndS (medium) = magnitude and direction of the slope in the scalar field at a point

Gradient Component of gradient is the partial derivative in the direction of that component Fourier´s Law of Heat Conduction

The integral definition given on a previous slide can also be used to obtain the formulas for the gradient. On the next four slides, the form of GradF in Cartesian coordinates is worked out directly from the integral definition.

Differential form of the Gradient Cartesian system Evaluate integral by expanding the variation in  about a point P at the center of an elemental Cartesian volume. Consider the two x faces:  = (x,y,z) k dz i P j adding these gives Face 2 Proceeding in the same way for y and z we get and , so Face 1 dx dy

. Gradient of a Differentiable Function, F An element of volume with a local Cartesian coordinate system having its origin at the centroid of the corners, O x y z Δx Δz Δy M Point M is at the centroid of the face perpendicular to the y-axis with coordinates (0, Δy/2, 0) Other points in this face have the coordinates (x, Δy/2, z) . O Gradient of a Differentiable Function, F continued

Differential Forms of the Gradient Cartesian Cylindrical Spherical These differential forms define the vector operator 

Divergence Fluid particle, coincident with  at time t, after time t has elapsed. n dS Elemental volume  with surface S = proportionate rate of change of volume of a fluid particle

Differential Forms of the Divergence Cartesian Cylindrical Spherical

Differential Forms of the Curl Cartesian Cylindrical Spherical Curl of the velocity vector V = twice the circumferentially averaged angular velocity of the flow around a point, or a fluid particle =Vorticity Ω Pure rotation No rotation Rotation

Curl Elemental volume  with surface S e n dS Perimeter Ce Area  dS=dsh radius a v avg. tangential velocity = twice the avg. angular velocity about e