EGN 3353C Fluid Mechanics

Transcription

1 Lecture 4 When nondiensionalizing an equation, nondiensional araeters often aear. Exale Consider an object falling due to gravity in a vacuu d z ays: (1) the conventional diensional aroach, and () diensionless aroach. Integrating tice and alying the initial elevation z and velocity 1 z z t gt = + diensional anser dt = g. We ill solve for the elevation z to If e ant to deterine z( t ), e need to secify z,, and g (3 araeters). If one of these changes, e have to reeat the calculation! Instead e can nondiensionalize the above equation by z to give No define soe ne diensionless variables: The last ter becoes 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 ter is a diensionless z araeter. It is related to the ell-knon oude nuber = in Fluid Dynaics. MAE Det.

2 So the solution can be ritten in diensional or nondiensional for as follos: 1 z z t gt = + or z 1 t = 1+ t * * Ignoring tie, nd equation is only a function of 1 araeter * =, hile the first is a function of 3! Key Point: If you had to coute a table of solutions z( t ), you ould have to ick several different values of z,, and g and then calculate z( t ). If you choose 5 different values for each, that s = 15 * * lots! On the other hand, you only have to lot z ( t ) for several different values of values of ). To calculate z( t ), just read convert t to t * and read = (5 different * z fro the aroriate lot (erhas using interolation) and then convert * z to z. o In suary, there are key advantages of nondiensionalization: 1. it increases our insight about the relationshis beteen key araeters = shos that doubling is equivalent to reducing g or z by a factor of four. it reduces the nuber of araeters in the roble 15 lots vs. 5! MAE Det.

3 Trajectories of a steel ball falling in a vacuu. The results are nondiensionalized. This lot ould even be valid for variable g. Trajectories of a steel ball falling in a vacuu: (a) fixed at 4 /s, and (b) z fixed at 1 (Exale 7 3). Here, g is fixed. MAE Det.

4 DIMENSIONAL ANALYSIS AND SIMILARITY o Engineering often requires exerients (nuerical siulations or hysical) on geoetrically scaled odels rather than full-scale rototyes o To do this roerly requires a tool called diensional analysis o Three riary uroses 1. To generate nondiensional araeters that hel in the design of exerients (hysical and/or nuerical) and in the reorting of exeriental results. To obtain scaling las so that rototye erforance can be redicted fro odel erforance 3. To (soeties) redict trends in the relationshi beteen araeters Underlying concet siilarity o Three necessary conditions ust be satisfied beteen a odel and a rototye 1. geoetric siilarity (odel and rototye have exactly the sae shae and differ only by a scale factor). kineatic siilarity (V at any oint differs only by a constant scale factor beteen odel and rototye) geoetric siilarity is a rerequisite for kineatic siilarity ilies that all kineatic features (strealines, vorticity, etc.) ill be identical in odel and rototye MAE Det.

5 3. dynaic siilarity (all F and M scale by a constant factor beteen odel and rototye) kineatic siilarity is a rerequisite for dynaic siilarity o In a general flo field, colete siilarity beteen a odel and rototye is achieved only hen there is dynaic siilarity (since dynaic siilarity ilies geoetric and kineatic siilarity) Diensionless Paraeters called Π araeters or grous o Usually ritten in folloing for Π = f Π, Π, Π,..., Π k deendent indeendent araeters araeter o total of k Π grous e ill sho a rocedure to deterine these Π grous o Don t kno functional for of f need exerients to deterine this in general o if dynaic siilarity is achieved, all Π s ill be equal beteen odel and rototye MAE Det.

6 Exale Aerodynaic drag on a car. Wind tunnel test at sae conditions (sae sae teerature). What is the velocity and force scale factors for dynaic siilarity? We ill see that there are only Π grous for incoressible flo over a car. FD Π 1 =, Π = V L drag coefficient VL μ Reynolds nuber Model ust be an exact relica (usually saller) of the rototye. Hence, Π =Π and Π =Π. 1 1 For 1/5 th scale odel for air ind tunnel (sae sae teerature): VL μ VL L μ = or V = V = 5V. μ L μ = 1 = 5 = 1 If V = 55 h, V = h! Might be too high and ight FD FD violate incoressible assution. Also, = V L V L or L V FD = F D = F D. L V = 1 = 5 = 1/5 MAE Det.