EGN 3353C Fluid Mechanics

Transcription

1 Chapter 7: DIMENSIONAL ANALYSIS AND MODELING Lecture 3 dimesio measure of a physical quatity ithout umerical values (e.g., legth) uit assigs a umber to that dimesio (e.g., meter) 7 fudametal dimesios from hich all other secodary dimesios ca be expressed i terms of o Fluid dyamics usually just cocered ith m, L, t, ad T.

2 Example: Write the dimesios of electrical voltage i terms of primary dimesios. Poer = Voltage Curret [ Voltage] = Poer [ Curret ] = mlt Lt I = ml t I force velocity La of Dimesioal Homogeeity Every additive term i a equatio must have the same dimesios o Each term must also have the same uits o Recommedatio: Write out all uits he performig calculatios i order to avoid errors. Example: Various Forms of Beroulli s Equatio p + ρv + ρ = P t static hydrostatic total pressure q = dyamic pressure pressure pressure total pressure form P = ForceArea= mlt L =ml t each term must have the same uits (e.g., Pa ) p 1 + V + = C eergy/mass form ρ PE/mass differet costat each term has primary dimesios of [ ] [ ] flo eergy/mass KE/mass for each streamlie each term has primary dimesios of each term must have the same uits (e.g., - - = mlt L m = L t m s )

3 p V z = H form elevatio total ρ g g Pressure velocity each term has primary dimesios of L each term must have the same uits (e.g., m ) Nodimesioalizatio of Equatios o If e divide each term i a equatio by a term that has the same dimesios, the equatio is redered odimesioal. Each additive term i the equatio is dimesioless. 1. Example: If e divide the total pressure form of Beroulli Equatio by P t, e get 1 ρv p ρ + + = 1 P P P t t t o If, i additio, the odimesioal terms i the equatio are of order uity, the equatio is called ormalized. Sometimes, oe or more terms may be small ad ca be eglected.. Example: Suppose e have a term i a equatio that has the form ( U u) + here it. U + u = U + u U = U + u U = U + U + ε ( 1 ) 1 ( 1 ε ) ( 1 ε ) ( ) ( ) if ε << 1. If 1 ε, ( U u) U +.

4 o I the process of odimesioalizig a equatio of motio, odimesioal parameters ofte appear. Example Cosider a object fallig due to gravity i a vacuum d z ays: (1) the covetioal dimesioal approach, ad () dimesioless approach. Itegratig tice ad applyig the iitial elevatio z ad velocity 1 z z t gt = + dimesioal aser dt = g. We ill solve for the elevatio z to If e at to determie z( t ), e eed to specify z,, ad g (3 parameters). If oe of these chages, e have to repeat the calculatio! Istead e ca odimesioalize the above equatio by z to give No defie some e dimesioless variables: The last term becomes z t 1 gt = 1+ z z z * ( ) z z t =, t = z z * * t 1 g t z 1 * = = t z 1 g. The boxed term is a dimesioless z parameter. It is related to the ell-ko oude umber = i Fluid Dyamics.

5 So the solutio ca be ritte i dimesioal or odimesioal form as follos: 1 z z t gt = + or The secod equatio is oly a fuctio of 1 parameter z 1 t = 1+ t * * * =, hile the first is a fuctio of 3! Key Poit: If you had to compute a table of solutios z( t ), you ould have to pick several differet values of z,, ad g ad the calculate z( t ). If you choose 5 differet values, that s 5 5 5= 15 plots! O * * the other had, you oly have to plot z ( t ) for several differet values of values of ). To calculate z( t ), just read covert t to t * ad read = (maybe 5 differet * z from the appropriate plot (perhaps usig iterpolatio) ad the covert * z to z. o I summary, there are key advatages of odimesioalizatio: 1. it icreases our isight about the relatioships betee key parameters = shos that doublig is equivalet to reducig g or z by a factor of four. it reduces the umber of parameters i the problem 15 plots vs. 5!

6 Trajectories of a steel ball fallig i a vacuum. The results are odimesioalized. This plot ould eve be valid for variable g. Trajectories of a steel ball fallig i a vacuum: (a) fixed at 4 m/s, ad (b) z fixed at 1 m (Example 7 3). Here, g is fixed.