TYPES OF FLUID FLOW CLASS

NOTES FOR ENGINEERS

There are different types of flows

for a moving fluid. The study of

moving fluids is quite complex,

tedious and time consuming. But

the analysis can be made simple by

making one or more simplifying

assumptions e.g., one-dimensional

flow, steady-state flow, non-viscous

flow, compressible and in-compressible flow.

Compressible flow is that branch of

fluid mechanics which deals with flow

of fluids having significant changes in

fluid density. Gases display such a

behavior since liquids are in-compressible.

 Mach number  (the ratio of the speed of

the fluid flow to the speed of sound)

decides about the compressible and

in-compressible flow. When Mach

number is greater than 0.3, it is a

compressible flow since as Mach

number (M) increases, the density

changes become significant.  This

flow is found in high-speed aircraft,

space-exploration vehicles, jet

engines, gas pipelines, modern aircraft,

missiles, and spacecraft. Compressible

fluid flow study is complex. To date its

analysis is empirical in nature and

hence it is based on experimental data

and practical experience. More are the

simplifying assumptions, more easy will

be analysis but we will be faraway from reality.

TYPES OF FLUID FLOW

(i) Steady and Unsteady flows
(ii) Uniform and Non-uniform flows
(iii) Laminar and Turbulent flows
(iv) Compressible and In-compressible flows
(v) Rotational and Ir rotational flows
(vi) One, Two and Three dimensional flows

Sr.No.	Type of Flow	Definition	Mathematical representation
1.	Uniform Flow	When properties of the fluid do not vary with distance (position),(When properties gradient is zero), flow is uniform. .	velocity gradient du / dx =0 pressure gradient dp / dx =0 viscosity gradient, dµ/ dx =0 density gradient dρ/dx =0 etc.
2.	Non- Uniform Flow	When properties of the fluid vary with distance (Position),(When properties gradient is not zero), flow is non uniform.	velocity gradient du / dx ≠0 pressure gradient dp / dx ≠0 viscosity gradient, dµ/ dx ≠0 density gradient dρ/dx ≠0 etc.
3.	Steady Flow	When properties of the fluid do not vary with time (rate of change),it is steady flow.	du / dt =0 i.e. rate of change =0 dp / dt =0 i.e. rate of change =0 dµ / dt =0 i.e. rate of change =0 dρ / dt =0 i.e. rate of change =0
4.	Unsteady Flow	When properties of the fluid vary with time, it is unsteady flow	du / dt ≠0  i.e. rate of change ≠0 dp / dt ≠0  i.e. rate of change ≠0 dµ / dt ≠0  i.e. rate of change ≠0 dρ /dt ≠0  i.e. rate of change ≠0
5.	Uniform Steady Flow	When properties of the fluid do not vary with time as well as with distance (position), it is called a Steady uniform flow.	du /dt =0 and du/dx=0 dp /dt =0 and dp/dx=0 dµ /dt =0 and dµ/dx=0 dρ/dt =0 and dρ/dx=0
6.	Non-Uniform Steady Flow	When properties change with position but do not change w. r. t. time, it is non uniform steady flow.	du /dt =0 and du/dx≠0 dp /dt =0 and dp/dx≠0 dµ /dt =0 and dµ/dx≠0 dρ/dt =0 and dρ/dx≠0
7.	Uniform un- Steady Flow	When properties do not change with position but change with time, it is uniform non steady flow.	du / dt ≠0, du / dx =0 dp / dt ≠0, dp / dx =0 dµ / dt ≠0, dµ/ dx =0 dρ /dt ≠0, dρ/dx =0
8.	Non-Uniform Non-Steady Flow	When properties change with position as well as with time, it is unsteady non uniform flow.	du /dt ≠0 and du/dx≠0 dp /dt ≠0 and dp/dx≠0 dµ /dt ≠0 and dµ/dx≠0 dρ /dt ≠0 and dρ/dx≠0

Sr. No.	Type of Flow	Definition	Application
1.	One Dimensional Flow	When two velocity components are negligible	fully-developed flows in long uniform pipes and open-channels in which velocity is uniform across the pipe cross-section
2.	2-Dimensional	When one velocity components is negligible	flow past a long structure / Flow over a long weir/Flow past an antenna
3.	3-Dimensional	When all the three velocity components are equally important	Actual flow anywhere and everywhere
4.	Compressible flow	When there is a change in density with pressure	d ρ/dp ≠0,  Applicable to flow of gases and vapors
5.	Incompressible flow	When there is a no change in density with pressure	d ρ/dp =0, Applicable to flow of liquids, Applicable  for gases when Mach No < 0.3 OR Velocity of gases is < 100 m/s
6.	Viscous flow	When viscosity effect is predominant	All fluids have significant viscosity except water, air and kerosene
7.	Non-viscous	When viscosity effect is negligible	Flow of water, air and kerosene
8.	External flow	When flow is on the outer boundary of an object	Flow external to a cylinder, sphere, aero-foil, flow in a boundary layer
9.	Internal Flow	When flow is within some boundary which may be circular, rectangular or any other cross section	Flow through a pipe/ tube, flow in a duct, Flow through hydraulic machines
10.	Laminar flow	Fluid layers are flowing parallel to each other	(i)  In a rough pipe,     Re < 2100 (ii) In a smooth pipe, Re <10000 (iii) Over a flat plate, Re < 5 Lac
11.	Turbulent Flow	Fluid layers are crossing to each other and are in disorder	(i)  In a rough pipe, Re > 4000                                                                                                                                                                                        (ii) In a smooth pipe, Re >10000                                                                                                                                                                                        (iii) Over a flat plate, Re >7 Lac
12.	Sub-critical flow	flow in a channel where gravity forces are predominant	Froude Number = Fr = V/(gD)0.5 V=mean velocity of flow when D = Hydraulic depth of channel D = Area of flow/Channel top width Fr < 1
13.	Critical flow	flow in a channel where gravity forces are predominant	Fr = 1
14.	Super critical flow	flow in a channel where gravity forces are predominant	Fr > 1

COMPRESSIBLE AND IN-COMPRESSIBLE FLOW

Salient features of a compressible flow

Ludwig Prandtl found the following features linked with the compressible flow.

1. Boundary layer

2. Supersonic shock waves

3.  wind tunnels have supersonic flow

4. Design of nozzles with supersonic flow.

TABLE: Comparison of Compressible and In-compressible Flow

Sr. No.	Compressible flow	In-compressible flow
1.	Density changes with pressure or temperature or with both	Density is constant.
2.	Found in vapors and gases	Found in liquids. Also found in gases also as long as Mach Number is ≤ 0.3
3.	Variables are velocity, pressure, density, temperature and change in entropy	Variables are only pressure and velocity
4.	Equations used are continuity, momentum, energy and equation of state, change of entropy	Equations used are continuity and momentum
5.	Continuity equation dρ/ρ + dA/A + dV/V =0,  or ρ1 A1 V1 =  ρ2 A2 V2	Continuity equation A1V1 = A2V2
6.	Momentum eq.  ∫dp/ρ + v2/2 =constant	Momentum eq. ∫dp/ρ + v2/2 =constant
7.	Energy eq. dp/ρ + V dV +g dz = 0                      p1 + ρV12 /2 = p2 + ρ V22/2 (Bernoulli’s Equation)	Not applicable
8.	Equation of state dp/p –dρ/ρ –dT/T =0	Not applicable
9.	Equation of change of entropy ds=Cp (dT/T) — R dp/p	Not applicable
10.	Analysis is complex, cumbersome and time consuming because of more number of unknowns.	Analysis is simple
11.	Mathematically ∂ρ/∂p ≠0	Mathematically  ∂ρ/∂p =0
12.	Best way to analyze is to use Mach number	Mach number is not of much use
13.	Mach number = velocity of flow/velocity of sound	Mach number = velocity of flow/velocity of sound
14.	M ≥ 0.3	M ≤ 0.3
15.	Study of compressible flow is called Gas Dynamics	Called as incompressible flow
16.	Formation of waves is there.	No waves formation
17.	Found in gas flow through nozzles, orifices, Venturi-meters, compressors, turbines, airplanes, projectiles, missiles, rockets, space air crafts, water hammer,	Mainly found in liquids. Also in gases with M ≤ 0.3
18.	The stagnation values (at zero velocity) p o, To, ρo and h0 are useful reference conditions in an ISENTROPIC compressible flow	Not applicable
19.	Po = p[1 + ((γ –1) /2)M2]((γ / (γ—1))	Not applicable
20.	To = T[1 + ((γ –1) /2)M2]	Not applicable
21.	ρo = ρ[1 + ((γ –1) /2)M2]((1 / (γ—1))	Not applicable
22.	ho = h + v2/2	Not applicable
23.	Uses both Lagrangian and Euler approaches and hence becomes very complex	Uses only Lagrangian approach and is thus simpler
24.	Mathematically dρ /ρ  ≥ 5 %	Mathematically dρ/ρ ≤ 5 %
25.	Compressibility effect is considered. Z = p/(ρ R gas T)	Compressibility effect is neglected

 

Methods to Study Gas Dynamics

1.  Model experiments in a wind tunnel

2. Shock tubes using optical techniques

Computational Fluid Dynamics

It analysis the compressible flow. It uses supercomputers to analyze a variety of geometries and flow characteristics in a compressible flow. Both internal and external compressible flows can be evaluated. Computational fluid dynamics is an inexpensive alternative to experimental studies.

Assumptions Used in Compressible Flow

1. Fluid flow as a continuous substance. There are no voids or impurities in it.

2. There is no-slip condition. In most cases, the velocity of solid surface is zero. Because of no slip condition, flow becomes viscous and a boundary layer is developed.

3. In an in-compressible fluid flow, pressure and velocity are two unknown parameters. These are solved by using two equations, the continuity and linear momentum conservation equations. In compressible flow, pressure, velocity, density and temperature are four unknown variables. This requires the use of two more equations i.e. the conservation of energy equation and the equation of state.

4. Compressible fluid dynamics uses both Lagrangian and Eulerian frame of references because of its complex nature. The Lagrangian approach follows a particular particle or a group of particles of fixed identity. The Eulerian reference frame uses a fixed control volume that fluid can flow through.

NORMAL SHOCK IN A COMPRESSIBLE FLUID FLOW

When the flow passes from a supersonic to subsonic in a SMALL distance, the velocity decreases suddenly and pressure rises sharply. This sudden pressure rise is a normal to the pipe surface and is called the NORMAL SHOCK. It happens only in a compressible flow (gas flow). The vice versa is not applicable i.e. there is no shock wave from subsonic to super- sonic flow. These shock waves are perpendicular to the flow. Shock waves are highly localized IR-REVERSIBILITIES in a compressible fluid (gas flow) flow. These are highly undesirable and should be avoided as far as possible.

Shock wave occurs when flow is changing from super-sonic to subsonic in a compressible flow (air/gas flow) over a very small (molecular) distance. There is a sudden increase in pressure, density, temperature and entropy when a shock is formed.
A shock wave is a moving disturbance. When a disturbance moves faster than the speed of sound in a fluid, it is a shock wave. A shock wave carries energy, and can propagate through a medium. It is characterized by an ABRUPT AND INSTANTANEOUS CHANGE in temperature, pressure, density, entropy, velocity and Mach number of the medium (fluid). It is normal to the flow direction and is thus called a normal shock.
A sound wave, similar to a shock wave, is heard as the familiar “thud” when a supersonic aircraft moves in the air. A shock wave is similar to a sound wave created by an object traveling through the air faster than the speed of sound. In smooth flight, the shock wave starts at the nose of the aircraft and ends at the tail. There is a rise in pressure at the front nose of the aircraft and pressure decreases steadily and becomes a negative pressure at the tail of the aircraft. Suddenly there is a return to normal pressure after the object passes. This is called Normal-wave.

Normal Shock, Fanno Line and Rayleigh Line in a Compressible Flow (Gas FLOW)

FANNO LINE

Fanno flow is the locus of points with the same mass flux (G=constant) and the same stagnation enthalpy (h⁰=constant). Stagnation means at zero velocity. Fanno lines are thus lines of CONSTANT STAGNATION TEMPERATURE, and hence of CONSTANT STAGNATION ENTHALPY:
Fanno flow is for one dimensional adiabatic flow (dq=0) in a duct of constant area with FRICTION.

RAYLEIGH LINE

Rayleigh line is a locus of points with same impulse pressure and same mass flux. It represents states of constant mass flux (flow per unit area) when heat is transferred to or from a gas (dq ≠0). It is a non- adiabatic flow line.

FANNO LINE AND RAYLEIGH LINE

Fanno lines and Rayleigh lines are often considered together. These lines are represented on enthalpy entropy chart. The intersection of Fanno line and Rayleigh line represents the end points in a normal shock for the same mass flux ‘G’. Each point on the Fanno line has a different Mach number.
Each point on the Rayleigh line has a different Mach number.
Even the sonic points ( M=1) are different on these lines. But these lines do intersect at two points. One of these intersecting point lies on supersonic region while other lies on subsonic region. Thus line joining these points represents the normal shock wave. FLOW WITH A CONSTANT AREA DUCT CAN SWITCH BETWEEN THE FANNO AND RAYLEIGH LINE.

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