MEE 307
Fluid Mechanics II
Md. Nur Alam Mondal
Assistant Professor
Dept. of Mechanical Engineering, HSTU
Content
• Dimensional analysis and similitude;
• Fundamental relations of compressible flow; Speed of sound wave;
• Stagnation states for the flow of an ideal gas;
• Flow through converging diverging nozzles; Normal shock.
• Real fluid flow; Frictional losses in pipes and fittings.
• Introduction to boundary layer theory; Estimation of boundary layer and momentum thickness,
Skin friction and drag of a flat plate.
• Introduction to open channel flow; Best hydraulic channel cross-sections; Hydraulic jump;
Specific energy; Critical depth.
Reference
• Fluid Mechanics- F.M. White
• Class Lectures
• Online help for more details
Dimensional analysis
❑ Dimensional analysis is a method for reducing the number and complexity of experimental
variables which affect a given physical phenomenon, by using a sort of compacting technique.
❑ Suppose one knew that the force F on a particular body immersed in a stream of fluid depended only on
the body length L, stream velocity V, fluid density ρ, and fluid viscosity μ, that is, F = f(L, V, ρ , μ )
Here, the value of F is very difficult in experimental set up. In this case, dimension analysis is the
only option
Benefits:
• enormous savings in time and money.
• it helps our thinking and planning for an experiment or theory
• provides scaling laws which can convert data from a cheap, small model to design information for an
expensive, large prototype.
Dimensional analysis
❑ If an equation truly expresses a proper relationship between variables in a physical process, it will be
dimensionally homogeneous; i.e., each of its additive terms will have the same dimensions.
The Choice of Variables and Scaling Parameters:
• The variables are the things which we wish to plot, the basic output of the experiment or theory
• The scaling parameters are those quantities whose effect upon the variables we wish to know.
Buckingham Pi Theorem
If a physical process satisfies the PDH (homogeneous equation) and involves n dimensional variables, it
can be reduced to a relation between only k dimensionless variables or ’s. The reduction j = n - k equals
the maximum number of variables which do not form a pi among themselves and is always less than or
equal to the number of dimensions describing the variables.
❑ If a phenomenon depends upon n dimensional variables, dimensional analysis will reduce the
problem to only k dimensionless variables, where the reduction n - k = 1, 2, 3, or 4, depending
upon the problem complexity.
❑ Generally n - k equals the number of different dimensions (sometimes called basic or primary
or fundamental dimensions) which govern the problem. In fluid mechanics, the four basic
dimensions are usually taken to be mass M, length L, time T, and temperature θ, or an MLTθ
system for short. Sometimes one uses an FLTθ system, with force F replacing mass.
The force F on a particular body immersed in a stream of fluid depended only on the body length L,
stream velocity V, fluid density ρ, and fluid viscosity μ, that is, F = f(L, V, ρ , μ )
Dimensionless Parameters
Inertia force= mass x acceleration= ρL3. L/T=ρU2L2
Viscous Force = area x shear stress = A. τ = L2. μ.U/L
Dimensionless Parameters
Inertia force= mass x
acceleration
= ρL3. L/T=ρU2L2
Viscous Force
= area x shear stress
= A. τ = L2. μ. U/L
Pressure force
= Pressure x area
=P. L2
Dimensionless Parameters
Inertia force= mass x
acceleration
= ρL3. L/T=ρU2L2
Gravity Force
= mass x acceleration
= ρL3 . g
Surface tension force
= surface tension x length
=ϒ. L
Dimensionless Parameters
Mach number is the square root of inertia force and elastic force.
Where,
Inertia force= mass x acceleration
= ρL3. L/T=ρU2L2
Elastic force = Elasticity x area = E. L2
= ρa2.L2
Assignment #01
Assignment #01
Assignment #01
