Lecture # 1: Course Introduction and Similitude of Experiments

AerE 545 - Lecture Notes

Dr. Hui Hu

Department of Aerospace Engineering Iowa State University Ames, Iowa 50011, U.S.A

• Syllabus • Course policy

• Material Properties: ρ,m, specific volume, µ ,γ ,D (Most of them can be found in handbooks)

• Kinematic Properties: Describes the fluid motion w/o considering the force. (Position, V, displacement, acceleration, momentum, volume flow rate, mass flow rate, etc)

• Dynamic properties: Related to applied forces. (Pressure, shear stress , Torque)

• Thermodynamic properties: Heat and Work. (T, e, h, S)

• Lagrangian Method Focused on fluid particles ∆L V = lim ∆t→0 ∆t

• Eulerian Method: Focused on space location.

U (xi ,t)= V (x0i ,t) Acceleration: r r DV a = ⇒ Langragian domain  Dtr r ∂U r r a = + (U •∇)U  ∂tr r r r  ∂U ∂U ∂U ∂U  = +U1 +U 2 +U3 ⇒ Eulerian domain  ∂t ∂x1 ∂x2 ∂x3   1  ∂U i ∂U i  eij =  +  Rate of Strain: 2  ∂x j ∂x j 

• Primary Properties: Properties which are independent to each other

Name Abbreviations Unit Length L M Mass m kg Time t s Temperature T K Electric current I A Amount of substance mole mol Luminous intensity Candela Cd Plane Angle Radius rad Solid Angle Storadian Sr

• Secondary Properties: Related to other properties through their definition or basic principles

• Similitude: • The study of predicting prototype conditions from model observations.

F-22 Raptor Air Superiority Fighter

∆pl = f (D, ρ, µ,V )

∆p ρVD = Φ( ) ρV 2 µ

• Step 1: List all the variables that are involved in the problem. • Step 2: Express each of the variables in terms of basic dimensions. – Basic dimension: M, L,T, F – Force - F=MLT -2, density - ρρρ=ML -3; or ρρρ=FL -3T2. • Step 3: Determine the required number of pi-terms. – Number of pi-terms is equal to k-r, where k is the number of vearibel in the problem, r is the number if reference dimensions required to described the variables. • Step 4: Select a number of repeating variables, where the number required is equal to the number of reference dimensions. • Step 5; Form a pi-term by multiplying one of the non-repeating variables by the product of repeating variables, each raised to an exponent that will make the combination dimensionless. • Step 6: Repeat Step 5 for each of the remaining non-repeating variables. • Step 7: Check all the resulting pi terms to make sure they are dimensionless • Step 8: Express the final form as a relationship among the pi-terms, and think about what it means. K Π1 = Φ(Π 2 ,Π ,3 Π k −r )

• Example −3 ∆pl = FL D = L −4 2 ∆pl = f (D, ρ, µ,V ) ρ = FL T V = LT −1 µ = FL −2T K = ;5 r = 3 ⇒ 2 π − terms is needed

a b c Π1 = ∆pl D V ρ

 1+ c = 0  a =1   ∆p D −3 a −1 b −4 2 c 0 0 0 ⇒ ⇒ ⇒ l (FL )( L) (LT ) (FL T ) = F T L − 3+ a + b − 4c = 0 b = −2 Π1 = 2   ρV  − b + 2c = 0 c = −1

a b c Π 2 = µD V ρ  1+ c = 0 a = −1   µ −2 a −1 b −4 2 c 0 0 0 ⇒ ⇒ (FL T )( L) (LT ) (FL T ) = F T L ⇒ − 2 + a + b − 4c = 0 b = −1 Π 2 =   DρV  1− b + 2c = 0 c = −1

V inertial force Mach Number, M = ∝ c compressib lity force ρVL inertial force Reynolds number, Re = ∝ µ viscous force ∆p pressure force Euler number, Eu = ∝ 1 ρV 2 inertial force 2 D Drag Drag Coefficien :t C = = D 1 ρV 2S inertial force 2 L Lift Lift Coefficien :t C = = L 1 ρV 2S inertial force 2 V momentum diffusion Prandtl Number : Pr = = γ heat diffusion U momentum Schmidt Number : Sc = = γ c mass V inertial force Froude Number, Fr = ∝ lg gravity force lϖ centrifuga l force Strohal Number, Str = ∝ V inertial force V 2lρ inertial force Weber Number, We = ∝ σ surface tension force L Copyright © by Dr. Hui Hu @ Iowa State University. All Rights Reserved! Similitude

• Geometric similarity: the model have the same shape as the prototype:

F-16 F-22

• Kinematic similarity: condition where the velcoity ratio is a constant between all corresponding points in the flow field. – The streamline pattern around the model is the same as that around the prototype

• Dynamic similarity: Forces which act on corresponding masses in the model flow and prototype flow are in the same ratio through out the entire flow. (F ) (F ) (F ) (F ) I m = p m = µ m = g m = constant (FI ) p (Fp ) p (Fµ ) p (Fg ) p (F ) (F ) (F ) (F ) ⇒ I m p m ⇒ I m I p ⇒ = = Eu m = Eu p (FI ) p (Fp ) p (Fp )m (Fp ) p (F ) (F ) (F ) (F ) ⇒ I m µ m ⇒ I m I p ⇒ = = Re m = Re p (FI ) p (Fµ ) p (Fµ )m (Fµ ) p

(FI )m (Fg )m (FI )m (FI ) p ⇒ = ⇒ = ⇒ Fr m = Fr p (FI ) p (Fg ) p (Fg )m (Fg ) p