Mechanical Engineering - Hydraulics and Fluid Mechanics - Discussion

In dimensional analysis we use dimensional numbers to check model similarity. In some physical situation elastic force become predominate like inertia force. Mach great scientist introduce a number, which is square root of ratio of inertia and elastic forces to determine similarity between model and prototype.

Here elastic force is product of bulk modulus and area. Thus we need k to determine mach number. And also inverse of mach no is cauchy's no, hence it also need value of k.

E = density*dp/change in density. From bernoulli's equation if the order of pressure is same as the order of dynamic pressure then dp=density*velocity square/2. By substituting this in equation for bulk modulus we get. Change in density/density=1/2* (mach. No) square. Where mach number Ma=v/a.

9 velocity of fluid flow/velocity of sound in medium. If change in density/density is < 0. 05 we say fluid is incompressible. So Ma, 0. 33 for the flow to be compressible.

The Mach Number is a dimensionless value useful for analyzing fluid flow dynamics problems where compressibility is a significant factor. (You can also read as-if we compare sound velocity to other velocity then it's called of mach number).

The Mach Number can be expressed as.

M = v/c (1).

Where,

M = Mach number.

v = Fluid flow speed (m/s, ft/s).

c = Speed of sound (m/s, ft/s).

Mach Number (M).
M<1 = subsonic flow.
M=1 = sonic flow.
M is 1 to 6 = supersonic flow.
M more than 6 = hypersonic flow.

Mach Number Or Cauchy's Number = Inertia force to elastic force.
Euler number = Inertia force to Pressure force.
Weber's Number = inertia force to surface tension force.
Froude's number = inertia to gravity force.
Reynold's number = inertia to viscous.

Alternatively, the Mach Number can be expressed with the density and the bulk modulus for elasticity as,

M = v (ρ/ E)1/2.

Where,

ρ = density of fluid (kg/m^3, lb/ft^3).
E = bulk modulus elasticity (N/m^2 (Pa), lbf/in^2 (psi)).

The bulk modulus elasticity has the dimension pressure and is commonly used to characterize the fluid compressibility.

Bulk modulus - (SOM), It is the ratio of volumetric stress to volumetric strain. In Fluid mechanics it is the ratio of compressive stress to volumetric strain. Both sentences are same. And compressibility is the reciprocal of bulk modulus of elasticity.

Mach Number can be expressed with the density and the bulk modulus for elasticity as;

M = v (ρ/ E)1/2.

where
ρ = density of fluid (kg/m3, lb/ft3).
E = bulk modulus elasticity (N/m2 (Pa), lbf/in2 (psi)).

Mach no = inertia force / elastic force . It is a dimensionless quantity also defined as ratio of velocity of the flow / velocity of sound. Velocity of sound = sqrt (y. R.T ) y - gama.

Bulk modulus is equal to the reciprocal of compressibility.

And Mach number is depends upon the compressibility.

Therefore to determine Mach number bulk modulus is required.

MACH NO OR CAUCHY'S NO IS the ratio of inertia force to elastic force & also ratio of speed of flow(v) to velocity of sound(c) so,

c= √K/row(p).
k is bulk modulus.