1、ES 202 Fluid and Thermal Systems Lab 1: Dimensional Analysis,Road Map of Lab 1,AnnouncementsGuidelines on write-upFundamental of dimensional analysis difference between “dimension” and “unit” primary (fundamental) versus secondary (derived) functional dependency of data Buckingham Pi Theorem alterna
2、tive way of data representation (reduction) active learning exercises,Announcements,Lab 2 will be at Olin 110 (4th week)Lab 3 will be at DL 205 (8th week)You are not required to hand in the in-class lab exercises.,About the Write-Up,Raw data sheet and write-up format for Lab 1 can be downloaded at h
3、ttp:/www.rose-hulman.edu/Class/me/ES202Due by 5 pm one week after the lab at my office (O-219),Dimension Versus Unit,Dimensions (units) Length (m, ft) Mass (kg, lbm) MLT system Time (sec, minute, hour) Force (N, lbf) FLT system Temperature (deg C, deg F, K, R) Current (Ampere),Primary Versus Seconda
4、ry,In the MLT system, the dimension of Force is derived from Newtons law of motion.In the FLT system, the dimension of Mass is derived likewise.Quantities like Pressure and Charge can be derived based on their respective definitions.Do exercises on Page 1 of Lab 1,Dimensional Homogeneity,The dimensi
5、on on both sides of any physically meaningful equation must be the same.Do exercises on Page 2 of Lab 1,Data Representation,Given a functional dependencyy = f (x1, x2, x3, ., xk )where y is the dependent variable while all the xis are the independent ones. Both y and the xis can be dimensional or di
6、mensionless. One way to express the functional dependency is to view the above relation as an n-dimensional problem: to plot the dependency of y against any one of the xis while keeping the remaining ones fixed.,Buckingham Pi Theorem,If an equation involving k variables is dimensionally homogeneous,
7、 it can be reduced to a relationship among k - r independent products (P groups), where r is the minimum number of reference dimensions required to describe the variables.,Alternative Way of Data Representation,It is advantageous to view the same functional dependency in a smaller dimensional spaceC
8、ast y = f (x1, x2, x3, ., xk-1 )into P 1 = g (P 2, P 3, P 4, ., P k-r )where P is are non-dimensional groups formed by combining y and the xis, and r is the number of reference dimensions building the xis,What is the Procedure?,Come up with the list of dependent and independent variables (the least
9、trivial part in my opinion)Identify the number of reference dimensions represented by this set of variables which gives the value of rChoose a set of r repeating variables (these r repeating variables should span all the reference dimensions in the problem)All the remaining k - r variables are autom
10、atically the non-repeating variables,Continuation of Procedure,Form each P group by forming product of one of the non-repeating variables and all the repeating variables raised to some unknown powers. For example,P = y x1a x2b x3cBy invoking dimensional homogeneity on both sides of the equation, the
11、 values of the unknown exponents can be foundRepeat the P group formulation for each of the non-repeating variables,Properties of P Groups,The P groups are not unique (depend on your choice of repeating variables)Any combinations of P groups can generate another P groupThe simpler P groups are the p
12、referred choices,Motivational Exercise,Drag on a tennis ball work out the whole problemwhat if it is not spherical, say oval?what if it is not placed parallel to flow direction but at an angle?,Any Advantages?,Absolutely “YES” You may reduce a thick pile of graphs to a single xy-plot For examples: 4
13、 variables in 3 dimensions can be reduced to 1 P group which is equal to a constant (dimensionless) 5 variables in 3 dimensions can be reduced to 2 P groups taking the general formP 1 = f (P 2 ),Drag Coefficient for a Sphere,taken from Figure 8.2 in Fluid Mechanics by Kundu,More Exercises,Sliding bl
14、ockPendulum,What is the Key Point?,There are more than one way to view the same physical problem.Some ways are more economical than othersThe reduction of dimensions from the physical dimensional variables to non-dimensional P groups is significant!,Reflection on the Procedures,The most important st
15、ep is to come up with the list of independent variables (Buckingham cannot help in this step!)Once the dependent and independent variables are determined (based on a combination of judgment, intuition and experience), the rest is just routine, i.e. finding the P groups!However, Buckingham cannot giv
16、e you the exact form of the functional dependency. It has to come from experiments, models or simulations.,Complete Similarity,Model versus prototype (full scale)Geometric similarityKinematic similarityDynamic similarity,Central Theme,The Dimensionless world is simpler!,More Examples,Sliding blockPendulumNuclear bombTerminal velocity of a falling objectPressure drop along a pipe,
