ME 3111 Chapter 5
The flow velocity is 285 m/s. The velocity of sound for a unit Mach number is
285
A physical process involves n dimensional variables and is reduced to a relation between k dimensional variables. If it satisfies the pi theorem, the maximum number of variables that do not form a pi among themselves is equal to _____.
n - k
Identify the parameters that do not have any fundamental units.
Counting numbers Pure constants
If Cp and Cv denote the specific heat at constant pressure and specific heat at constant volume respectively, the specific-heat ratio is given by _____.
Cp/Cv
A model of a prototype is developed on a scale 10:1. The pi constant for the model and the prototype is given by πm and πp, respectively. Identify the relation between πm and πp if similarity exists between the prototype and the model.
πm = πp
Continuity equation states that ∇⋅V = 0 and ∇*⋅V* = 0 represents continuity equation in a dimensionless form. Identify the relevant dimensionless variables if U and L represent the reference velocity and reference length.
∇* = L∇ V* = VU
Identify the dimensionless continuity equation of motion.
∇*⋅V* = 0
Identify the significance of the Froude number.
It is used in open-channel flow analysis.
Identify the true statements about dimensional analysis.
It suggests dimensionless ways of writing equations. It provides scaling laws that converts data from a cheap, small model to design an expensive, large prototype.
Identify the true statements that ensure the dynamic similarity for compressible flow.
The model and prototype should be geometrically similar. The Reynolds number of the model and the prototype are equal. The Mach number of the model and the prototype are equal.
If a prototype has a 10-mm boundary layer trip wire 2.5 m from the leading edge, the model should have a _____ mm trip wire 0.25 m from its leading edge.
1 (The prototype and the model are in a ratio of 10:1. The model should have a 1 mm trip wire 0.25 m from its leading edge.)
Rank the six main steps of the pi theorem in the order of their occurrence.
1. List the variables in the problem 2. List the primary dimensions of the variables 3. Set the reduction factor 4. Select the scaling parameters 5. Determine all the desired pi products 6. Ensure all the pi groups are ensured to be dimensionless
After integrating the function, ∫t dt, the pure constants that result are _____.
1/2 and exponential 2
A planet makes 400 revolutions about the sun. The distance between the planet and the sun is 10.5 light-years. The maximum distance recorded between the sun and the planet is 10.5 × 105 m. The acceleration due to gravity in the planet is 12.2 m/s2. Identify the dimensionless term from the following. Multiple choice question. 10.5 light years
400 revolutions
A pendulum is used to determine the acceleration due to gravity. Which of the following statements involves a dimensionless term?
A bob pendulum is displaced by 12 rad.
Which of the following statements has a dimensionless term?
A coin spins for 5 times about its axis.
Identify the true statement(s) about geometric similarity.
All the angles are preserved in geometric similarity. Surface roughness of the model is subjected to scaling in geometric modeling.
Identify the assumptions that are made before dimensional analysis is carried out for an equation.
All the relevant variables have been included in the proposed relation. The proposed physical relation is dimensionally homogeneous.
Identify a true statement of nondimensionalization of an equation.
All the terms in a nondimensional equation are dimensionless.
Which of the following methods is used to reduce the number and complexity involved in solving an equation?
Dimensional analysis
Identify the true statements about the application of the pi theorem.
Each pi group obtained is independent. Each desired pi group will be a power product of variables that do not form a pi group plus one additional variable. The additional variables added to the scaling parameters are assigned nonzero exponent values.
The dimensionless number that determines the cavitation in a liquid is the _____.
Euler number
True or false: Dimensional variables cannot be nondimensionalized as a dimensional analysis technique.
False
Identify examples of systems that are kinematically similar.
Flows with no free surfaces with independent time and length scale ratios Frictionless low-speed flows
Continuity equation states that ∇⋅V = 0 and Navier-Stokes equation states that ρdVdtdVdt = ρg - ∇p + μ∇2V. Using the concept of nondimensionalization, identify the boundary conditions for the two equations.
For solid surface: V = 0 Inlet or outlet: Known V, p
_____ number is defined as the ratio of buoyancy and viscosity.
Grashof
________ similarity exists simultaneously with dynamic similarity if the model and prototype force and pressure coefficients are identical.
Kinematic
If principle of dimensional homogeneity holds true for the equation pρpρ + 1212V2 + gz = constant, identify the dimensions of each additive term.
L2T-2
Identify the preferable scaling parameters.
Length Time Mass
The torque required to move object 1 is given by 'T', which has a dimension of ML^2/T^2. Thrice the torque 'T' is required to move object 2. The dimension of torque required by object 2 is _____.
ML^2/T^2
How can the counting numbers be represented in terms of fundamental dimensions?
M^0L^0T^0
The ratio of fluid speed to the sound speed gives the dimensionless ________ number.
Mach
Identify the factor that determines the similarity between a model and a prototype.
Pi functions
The force (F) acting on a ship depends on its length, flow velocity, density, and dynamic viscosity of water. F = f(L, V, ρ, μ) If the Buckingham's pi theorem is used to perform dimensional analysis and reduce the equation, the dimensionless force coefficient F/(ρV2L2) is found to be a function of the dimensionless ________ number.
Reynolds
Identify the guidelines that must be followed while selecting the repeating parameters.
Select simple and popular scaling variables. Do not select output variables for the scaling variables.
Identify the true statements about the Reynolds number.
The Reynolds number is used in fluid flow analysis. The Reynolds number relates the inertia force and the viscous force. The Reynolds number is a dimensionless parameter.
Identify the true statements about the Weber number.
The Weber number has a significant importance when the surface curvature is comparable in size to the liquid depth. If the Weber number is large, its effect can be neglected. The Weber number helps in the analysis of very small hydraulic models.
Identify the true statements about dimensional constants.
They can conceivably be nondimensionalised. They are used to nondimensionalize other variables
True or false: Each pi group obtained using the pi theorem should be dimensionless.
True
True or false: Flow conditions for a model test are completely similar if all relevant dimensionless parameters have the same corresponding values for the model and the prototype.
True
True or false: If an equation truly expresses a proper relationship between variables in a physical process, it will be dimensionally homogeneous.
True
True or false: The motions of two systems are kinematically similar if homologous particles lie at homologous points at homologous times.
True
Which of the following dimensionless numbers is used in the analysis of droplets and capillary flows?
Weber number
The Euler number has a significant importance when _____.
cavitation occurs in a liquid
The parameters that have dimensions, and may vary from case to case, but are held constant during a given run are called _____.
dimensional constants
The dimensional parameters that vary from case to case and are plotted against each other are called _____.
dimensional variables
When a model and its prototype have the same length scale ratio, time scale ratio, and force scale ratio, it satisfies the __________ similarity condition.
dynamic
The skin friction coefficient is the ratio of wall shear stress to _____.
dynamic pressure
Identify the pure constants from the following terms.
e (Exponential power)
The Froude number is the ratio of inertia force to _____ force.
gravity
