AOE 5104 Class 2 Online presentations: Homework 1, due in class 9/4 Fundamentals Algebra and Calculus 1 Homework 1, due in class 9/4 Grading Policy Study Groups Recitation times (recitations to start week of 9/8) Monday 5-6, 5:30-6:30… Tuesday 5-6, 5:30-6:30…

Streamline: Line everywhere tangent to the velocity vector 3a. Ideal Flow Viscous and compressible effects small (large Re, low M). Flow is a balance between inertia and pressure forces, i.e. acceleration vector balances the pressure gradient vector Acceleration vector Pressure gradient vector Streamline: Line everywhere tangent to the velocity vector

This image shows a single time step from an unsteady flow simulation over the Space Shuttle Launch Vehicle during launch configuration. The momentum is shown as streamlines colored by magnitude and the density on the shuttle surface is shown as filled color- mapped contours. The flow fields are computed over nine overlapping grids with almost 950,000 points. The data are available courtesy of NASA/Ames Research Center, Moffett Field, CA http://www.opendx.org

3b. Viscous Flow Importance of viscous effects governed by Re=1000000 Boundary layer: Thin region adjacent to a solid surface where friction slows the flow. There is no pressure gradient across a boundary layer No-slip condition: Fluid immediately adjacent to a solid surface does not move relative to it

3b Viscous Flow Viscous region not always confined to a thin layer Separation: Large region of viscous flow produced when the boundary layer leaves a surface because of an adverse pressure gradient, or a sharp corner

3c. Compressibility Incompressible Regime M<0.3 Importance of compressibility effects governed by Incompressible Regime M<0.3 Negligible compressibility effects Subsonic Regime 0.3<M<0.7 Quantitative effects, no qualitative effects Transonic Regime 0.7<M<1.3 Large regions of subsonic and supersonic flow. Large qualitative effects. Supersonic Regime M>1.3 Almost entirely supersonic flow. Large qualitative effects

Flow Past a Circular Cylinder Re = 10,000 and Mach approximately zero Re = 110,000 and Mach = 0.45 Re = 1.35 M and Mach = 0.64 Pictures are from “An Album of Fluid Motion” by Van Dyke

Flow Past a Circular Cylinder Mach = 0.80 Mach = 0.90 Mach = 0.95 Mach = 0.98 Pictures are from “An Album of Fluid Motion” by Van Dyke

Flow Past a Sphere Mach = 1.53 Mach = 4.01 Pictures are from “An Album of Fluid Motion” by Van Dyke

3c. Compressibility Some Qualitative Effects Shock wave: Very strong, thin wave, propagating supersonically, producing almost instantaneous compression of the flow, and increase in pressure and temperature. Hypersonic vehicle re-entry NASA Image Library

3c. Compressibility Expansion or isentropic compression wave Some Qualitative Effects Expansion or isentropic compression wave Finite wave (often focused on a corner), moving at the sound speed, producing gradual compression or expansion of a flow (and raising or lowering of the temperature and pressure). Cone-cylinder in supersonic free flight, Mach = 1.84. Picture from “An Album of Fluid Motion” by Van Dyke.

Summary What a fluid is. Its properties. The governing laws Reynolds number. Mach number How Newton’s 2nd Law works as a vector equation Viscous effects: no-slip condition, boundary layer, separation, wake, turbulence, laminar Compressibility effects: Regimes, shock waves, isentropic waves. Initial ideas of concepts such as streamlines/eddies Qualitative understanding

2. Vector Algebra

Vector basics Vector: A, A Magnitude: |A|, A Scalar: p,  Types Polar vector Velocity V, force F, pressure gradient p Axial vector Angular velocity , Vorticity Ω, Area A Unit vector i, j, k, es, n, A/A … Q MAG DIR P

Vector Algebra Addition A + B = C Dot, or scalar, product A.B = ABcos E.g. Work=F.s Flow rate through dA=V.dA or V.ndA A.B=B.A A.A=A2 A.B=0 if perpendicular A C A  B

Vector Algebra Cross, or vector, product AxB=ABsine AxB=-BxA AxA=0 AxB=0 if A and B parallel A  B Parallelogram area is |AxB| Measured to be <180o Perpendicular to A and B in direction given by RH rule rotation from A to B

Vector Algebra – Triple Products (A.B)C = (B.A)C Mixed product A.BxC Volume of parallelepiped bordered by A, B, C May be cyclically permuted A.BxC=C.AxB=B.CxA Acyclic permutation changes sign A.BxC=-B.AxC etc. Vector triple product Ax(BxC) = Vector in plane of B and C = (A.C)B – (A.B)C B C A BxC

PIV of Flow Downstream of a Circular Cylinder Chiang Shih , Florida State University

Cartesian Coordinates Coordinates x, y , z Unit vectors i, j, k (in directions of increasing coordinates) are constant Position vector r = x i + y j + z k Vector components F = Fx i+Fy j+Fz k = (F.i) i+ (F.j) j+ (F.k) k Components same regardless of location of vector z F k j i r z y x y x

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 F ez e er R z  r

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 er e F r e  r 

Vector Algebra in Components …works for any orthogonal coordinate system!

CFD of Flow Around a Fighter FinFlo Ltd

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