Gas Dynamics: Potential Flow Equation:

In fluid dynamics, potential flow describes the velocity field as the gradient of a scalar function: the velocity potential. As a result, a potential flow is characterized by an irrotational velocity field, which is a valid approximation for several applications. The irrotationality of a potential flow is due to the curl of the gradient of a scalar always being equal to zero.

In the case of an incompressible flow the velocity potential satisfies Laplace's equation, and potential theory is applicable. However, potential flows also have been used to describe compressible flows. The potential flow approach occurs in the modeling of both stationary as well as nonstationary flows.

Applications of potential flow are for instance: the outer flow field for aerofoils, water waves, electroosmotic flow, and groundwater flow. For flows (or parts thereof) with strong vorticity effects, the potential flow approximation is not applicable.

Gas Dynamics: Rayleigh flow

Rayleigh flow refers to frictionless, non-Adiabatic flow through a constant area duct where the effect of heat addition or rejection is considered. Compressibility effects often come into consideration, although the Rayleigh flow model certainly also applies to incompressible flow. For this model, the duct area remains constant and no mass is added within the duct. Therefore, unlike Fanno flow, the stagnation temperature is a variable. The heat addition causes a decrease in stagnation pressure, which is known as the Rayleigh effect and is critical in the design of combustion systems. Heat addition will cause both supersonic and subsonic Mach numbers to approach Mach 1, resulting in choked flow. Conversely, heat rejection decreases a subsonic Mach number and increases a supersonic Mach number along the duct. It can be shown that for calorically perfect flows the maximum entropy occurs at M = 1. Rayleigh flow is named after John Strutt, 3rd Baron Rayleigh. Click here for notes of Rayleigh Flow

Gas Dynamics: Fanno Flow

Fanno flow refers to adiabatic flow through a constant area duct where the effect of friction is considered. Compressibility effects often come into consideration, although the Fanno flow model certainly also applies to incompressible flow. For this model, the duct area remains constant, the flow is assumed to be steady and one-dimensional, and no mass is added within the duct. The Fanno flow model is considered an irreversible process due to viscous effects. The viscous friction causes the flow properties to change along the duct. The frictional effect is modeled as a shear stress at the wall acting on the fluid with uniform properties over any cross section of the duct.

For a flow with an upstream Mach number greater than 1.0 in a sufficiently long enough duct, deceleration occurs and the flow can become choked. On the other hand, for a flow with an upstream Mach number less than 1.0, acceleration occurs and the flow can become choked in a sufficiently long duct. It can be shown that for flow of calorically perfect gas the maximum entropy occurs at M = 1.0. Fanno flow is named after Gino Girolamo Fanno.

Refer Below Folder For Notes of Fanno Flow Click Here: Fanno Flow

Gas Dynamics: Reflection of Shocks, Intersection of Shocks

Topics covered are: 1. Reflection of shocks
2. Intersection of Shocks
2.1 Different Family(objects)
2.2 Same Family(objects)
3. Wave reflection from free Boundary

Gas Dynamics: Expansion Hodograph

Gas Dynamics: Flow over a wedge, Symmetrical wedge, Unsymmetrical wedge, Strong Shock, Weak Shock and detatched Shock

The below notes contains: Flow over a wedge, Symmetrical wedge, unsymmetrical wedge, Strong Shock, Weak Shock, Detached Shock

Gas Dynamics: Shock Polar

The term shock polar is generally used with the graphical representation of the Rankine-Hugoniot equations in either the hodograph plane or the pressure ratio-flow deflection angle plane. The polar itself is the locus of all possible states after an oblique shock.

Gas Dynamics: Oblique Shock Beta and Theta Relationship

Gas Dynamics: Oblique Shock Relations

An oblique shock wave, unlike a normal shock, is inclined with respect to the incident upstream flow direction. It will occur when a supersonic flow encounters a corner that effectively turns the flow into itself and compresses. The upstream streamlines are uniformly deflected after the shock wave. The most common way to produce an oblique shock wave is to place a wedge into supersonic, compressible flow. Similar to a normal shock wave, the oblique shock wave consists of a very thin region across which nearly discontinuous changes in the thermodynamic properties of a gas occur. While the upstream and downstream flow directions are unchanged across a normal shock, they are different for flow across an oblique shock wave.

Gas Dynamics: Pitot Static Tube Relations Derivation

A pitot tube is a pressure measurement instrument used to measure fluid flow velocity. The pitot tube was invented by the French engineer Henri Pitot in the early 18th century and was modified to its modern form in the mid-19th century by French scientist Henry Darcy. It is widely used to determine the airspeed of an aircraft, water speed of a boat, and to measure liquid, air and gas flow velocities in industrial applications. The pitot tube is used to measure the local flow velocity at a given point in the flow stream and not the average flow velocity in the pipe or conduit.

The basic pitot tube consists of a tube pointing directly into the fluid flow. As this tube contains fluid, a pressure can be measured; the moving fluid is brought to rest (stagnates) as there is no outlet to allow flow to continue. This pressure is the stagnation pressure of the fluid, also known as the total pressure or (particularly in aviation) the pitot pressure.

The measured stagnation pressure cannot itself be used to determine the fluid flow velocity (airspeed in aviation). However, Bernoulli's equation states:

    Stagnation pressure = static pressure + dynamic pressure 

Rankine Hugoniot Relation

The Rankine–Hugoniot conditions, also referred to as Rankine–Hugoniot jump conditions or Rankine–Hugoniot relations, describe the relationship between the states on both sides of a shock wave in a one-dimensional flow in fluids or a one-dimensional deformation in solids. They are named in recognition of the work carried out by Scottish engineer and physicist William John Macquorn Rankine and French engineer Pierre Henri Hugoniot.

Gas Dynamics Lecture Notes: Normal Shock

In elementary fluid mechanics utilizing ideal gases, a shock wave is treated as a discontinuity where entropy increases over a nearly infinitesimal region. Since no fluid flow is discontinuous, a control volume is established around the shock wave, with the control surfaces that bound this volume parallel to the shock wave (with one surface on the pre-shock side of the fluid medium and one on the post-shock side). The two surfaces are separated by a very small depth such that the shock itself is entirely contained between them. At such control surfaces, momentum, mass flux and energy are constant; within combustion, detonations can be modelled as heat introduction across a shock wave. It is assumed the system is adiabatic (no heat exits or enters the system) and no work is being done. The Rankine–Hugoniot conditions arise from these considerations.

Taking into account the established assumptions, in a system where the downstream properties are becoming subsonic: the upstream and downstream flow properties of the fluid are considered isentropic. Since the total amount of energy within the system is constant, the stagnation enthalpy remains constant over both regions. Though, entropy is increasing; this must be accounted for by a drop in stagnation pressure of the downstream fluid.

Gas Dynamics: Introduction to Shocks: Normal, Oblique shocks and Expansion Wave

 Shock is an abrupt discontinuity in the flow field. It occurs in flows when the local flow speed exceeds the local sound speed. More specifically, it is a flow whose Mach number exceeds 1.

Gas Dynamics - Flow through Nozzle derivation

The flow through a nozzle can be considered as a isotropic process. The isotropic process is the adiabatic reversible process.
No heat exchange between the system and surroundings.
No loss(frictional) within the system or with surroundings.

Gas Dynamics: C-D nozzle de Laval Nozzle

A de Laval nozzle (or convergent-divergent nozzle, CD nozzle or con-di nozzle) is a tube that is pinched in the middle, making a carefully balanced, asymmetric hourglass shape. It is used to accelerate a hot, pressurized gas passing through it to a higher speed in the axial (thrust) direction, by converting the heat energy of the flow into kinetic energy. Because of this, the nozzle is widely used in some types of steam turbines and rocket engine nozzles. It also sees use in supersonic jet engines.

Gas Dynamics: Area Vs Velocity Relationship Nozzle

To analyze the effect of change in stream tube cross sectional area on the characteristics of flow we consider:
1. Euler's equation for steady state one dimensional flow
2. Continuity Equation

Gas Dynamics Velocity of Sound

   Sound Speed represents the speed at which the medium transmits pressure disturbance. It also can be mentioned that the speed at which the particles transmit it's disturbance to the neighboring particle.

Gas Dynamics Continuity Equation Derivation

A continuity equation in physics is an equation that describes the transport of a conserved quantity. Since mass, energy, momentum, electric charge and other natural quantities are conserved under their respective appropriate conditions, a variety of physical phenomena may be described using continuity equations.

Continuity equations are a stronger, local form of conservation laws. For example, the law of conservation of energy states that energy can neither be created nor destroyed—i.e., the total amount of energy is fixed. But this statement does not immediately rule out the possibility that energy could disappear from a field in Canada while simultaneously appearing in a room in Indonesia. A stronger statement is that energy is locally conserved: Energy can neither be created nor destroyed, nor can it "teleport" from one place to another—it can only move by a continuous flow. A continuity equation is the mathematical way to express this kind of statement.

Gas Dynamics : Momentum Equation

Momentum of an object / praticle can be mentioned in two ways - Linear momentum and Angular Momentum.
The derivation of the momentum equation is given below.

Gas Dynamics Energy Equation derivation pdf.

The derivation of Energy equation is discussed in this lecture note.


Click here to download Energy equation lecture note.

Gas Dynamics - Introduction Lecture notes pdf

Compressible flow (gas dynamics) is the branch of fluid mechanics that deals with flows having significant changes in fluid density. Gases, but not liquids, display such behaviour.To distinguish between compressible and incompressible flow in gases, the Mach number (the ratio of the speed of the flow to the speed of sound) must be greater than about 0.3 (since the density change is greater than 5%) before significant compressibility occurs. The study of compressible flow is relevant to high-speed aircraft, jet engines, gas pipelines, commercial applications such as abrasive blasting, and many other fields.

Below is the sample of the lecture note and below that the link is given for you to download. All the notes will be uploaded in subsequent posts.

 

Click here to download the Introduction to Gas Dynamics

Lift, Drag, Pitching Moment and Center of Pressure of Supersonic Profiles pdf

Derivation of Lift, Drag, Pitching Moment and Center of Pressure of Supersonic Profiles:

Supersonic flow over a flat plate pdf

   When a fluid flow at the speed of sound over a thin sharp flat plate over the leading edge at low incident angle at low Reynolds Number. Then a laminar boundary layer will be developed at the leading edge of the plate. And as there are viscous boundary layer, the plate will have a fictitious boundary layer so that a curved induced shock wave will be generated at the leading edge of the plate. The shock layer is the region between the plate surface and the boundary layer. This shock layer be further subdivided into layer of viscid and inviscid flow, according to the values of Mach number, Reynolds Number and Surface Temperature. However, if the entire layer is viscous, it is called as merged shock layer.

Click here for lecture notes on Supersonic flow over a flat plate

Prandtl-Glavert Transformation for Subsonic Flow

The Prandtl–Glauert transformation is a mathematical technique which allows solving certain compressible flow problems by incompressible-flow calculation methods. It also allows applying incompressible-flow data to compressible-flow cases.


Click here for Lecture Notes of Prandtl-Glauert Transformation.

Small Perturbation Potential Theory

In Aerodynamics we are interested in the perturbation of a know fluid motion in the uniform steady flow.

A linearized differential equation for two dimensional potential flow with small perturbation, where the flow is incompressible is derived.

Click here for Small Protuberance Lecture Notes

Derivation of Equation of Potential Flow

 In fluid dynamics, potential flow describes the velocity field as the gradient of a scalar function: the velocity potential. As a result, a potential flow is characterized by an irrotational velocity field, which is a valid approximation for several applications. The irrotationality of a potential flow is due to the curl of the gradient of a scalar always being equal to zero.

 In the case of an incompressible flow the velocity potential satisfies Laplace's equation, and potential theory is applicable. However, potential flows also have been used to describe compressible flows. The potential flow approach occurs in the modeling of both stationary as well as nonstationary flows.

 Applications of potential flow are for instance: the outer flow field for aerofoils, water waves, electroosmotic flow, and groundwater flow. For flows (or parts thereof) with strong vorticity effects, the potential flow approximation is not applicable.


Click here for Derivation of Equation of Potential Flow Lecture Notes

CFD - Derivation of Momentum Equation, Navier Strokes, Spading Equation