# Oblique Shock Calculator – Properties of Oblique Shock Wave

This oblique shock calculator will help determine the **fluid flow properties** for an oblique shock wave. A shock wave is most commonly **associated with fast military aircraft** or spacecraft when they move faster than the **speed of sound**. The design of **air intakes for the aircraft** is based on calculating the desired fluid flow properties. The **air-breathing engines in military jets need air to enter the engine at subsonic speeds** to ensure the jet engines operate correctly.

This requirement is **achieved by introducing wedge-shaped objects** in the air takes to **compress the airflow before it reaches the combustion chamber**. You see some aircraft with **adjustable inlet cones that move axially** to increase or decrease the capture area, while others have **inlet ramps** for similar purposes.

The most common aircraft with **inlet cones** are the **Mikoyan Gurevich MiG-21, Lockheed F-104, and Dassault Mirage 2000**. Some aircraft like **Concorde, Lockheed Martin F22, and Mikoyan Gurevich MiG-25** use an **inlet ramp** instead of the cone to control the airflow. At supersonic speeds, the **air passing through these diffusers slows down due to the formation of shock waves** and enters the engine at slower speeds.

In addition to the desirable effects of the shock wave, there are also some **undesirable effects** such as **sonic booms** and **high-pressure wavefronts in a blast wave arising from an explosion** that can cause the loss of human lives. To this end, the underlying physics demands you to understand what is a shock wave and what are oblique shock relations. Read on to know how this oblique shock calculator can help you determine the fluid flow properties.

## What is a shock wave? — Normal and oblique shock waves

A shock wave is an **abrupt discontinuity** that **causes a change in fluid pressure, temperature, and density.** It is formed **when a wavefront travels at supersonic speed** and high pressure gets built up. This high-pressure wave travels faster than the local speed of sound and is heard as a **loud crack or whip** that is **produced by supersonic aircraft, explosions, and lightning strikes**.

Studies indicate a **shock wave has a thickness of about 200 nm**; therefore, we can consider it a

**line or plane**. The properties of the fluid —

**pressure, temperature, density, and velocity changed abruptly past this line**. These shocks are often forced by the

**use of different geometries**in an aircraft to achieve favorable post-shock conditions. The two common types of shock waves are:

- Normal shock wave; and
- Oblique shock wave.

When the shock wave is **deviating at an angle**, it is known as an *oblique shock wave*, whereas when the shock is **normal to the local flow**, it is called the *normal shock wave*.

## Oblique shock relations

As we know that the properties of flow change abruptly, let's look at the relations that one can use to define the pre (upstream) and post-shock (downstream) conditions. The **subscripts 1 and 2 represent upstream and downstream conditions**, respectively. Consider an

**airflow of pressure $p$, temperature $T$, and density $\rho$ and traveling at a Mach number, $M$.**The downstream Mach number is given by:

where:

- $\gamma$ – Specific heat ratio;
- $\beta$ – Wave angle; and
- $\theta$ – Turn angle.

The above equation is rewritten to eliminate the turn angle, $\theta$, such that:

You can obtain the Mach numbers for normal shock wave from respective oblique shock Mach numbers such that:

where:

- $M_x$ – Upstream Mach number for normal shock; and
- $M_y$ – Downstream Mach number for normal shock.

**Pressure ratio:** The relationship between upstream and downstream pressure is:

**Stagnation pressure ratio:** The stagnation conditions for upstream and downstream can be calculated using the Mach number and wave angle. Such that:

**Temperature ratio:** One can obtain the downstream temperature using the relation:

**Density ratio:** The ratio of densities at upstream to downstream is:

We can also write the stagnation pressure, density, and static pressure as:

This oblique shock calculator uses the above set of equations to determine the following upstream and downstream parameters:

- Mach numbers;
- Static pressure;
- Stagnation pressure;
- Temperature; and
- Density.

## How to calculate oblique shock wave properties

To find the oblique shock wave properties:

- Enter the
**upstream Mach number**, $M_1$. - Fill in the
**wave angle**, $\beta$. - The oblique shock calculator return the following properties:
**Upstream Mach number for the normal shock wave**, $M_x$;**Turn angle**, $\theta$;**Ratios**for pressure, temperature, density, and stagnation pressure;**Downstream Mach number for the oblique shock wave**, $M_2$; and**Downstream Mach number for the normal shock wave**, $M_y$.

**Flow properties**

If you have the **upstream properties**, you can use the `advanced mode`

of this calculator to find the values of downstream pressure, temperature, density, and stagnation pressure, or vice versa.

## Example: Using the oblique shock angle calculator

Find the pressure, temperature, and density ratio for an oblique shock wave having a wave angle of $20^\circ$ and upstream Mach number, `5`

.

To find the oblique shock wave properties:

- Enter the
**upstream Mach number**, $M_1 = 5$. - Fill in the
**wave angle**, $\beta = 20^\circ$. - Using the oblique shock relations:
**Turn angle**, $\theta = 10.664^\circ$;**Pressure ratio**, $p_2/p_1 = 3.245$;**Density ratio**, $\rho_2/\rho_1 = 2.214$; and**Temperature ratio**, $T_2/T_1 = 1.465$.

## FAQ

### What is an oblique shock wave?

The shock wave developed from the supersonic flow **inclined to the local fluid flow** is known as the oblique shock wave. This phenomenon results in a decrease of stagnation pressure and increases in entropy of the system. It has both desirable and undesirable effects.

### What is a normal shock wave?

When the developed shock wave is **perpendicular to the direction of the local fluid flow**, it is known as the normal shock wave. The shock wave is considered as a very thin line across which the velocity of the flow decreases and so does the downstream Mach number.

### How do I calculate pressure ratio for oblique shock wave?

To obtain the pressure ratio:

**Multiply the upstream Mach number**by the**sine of the wave angle**.- Find the
**square**of the**product**in the previous step. **Subtract**`1`

from the square.**Multiply**it by`2`

and**specific heat ratio.****Divide**the product by the**sum of specific heat ratio and**to obtain the pressure ratio.`1`

### How do I calculate density ratio for oblique shock wave?

To obtain the density ratio:

**Multiply the upstream Mach number**by the**sine of the wave angle**.- Find the
**square**of the**product**in the previous step. **Multiply**the square with the sum of specific heat ratio and`1`

to obtain the**numerator.****Subtract**`1`

from the**specific heat ratio**.**Multiply**the difference with the**square obtained in step**.`2`

**Add**`2`

to the**product**to obtain the**denominator.****Divide**the**numerator in step**with the`3`

**denominator in step**to obtain the`6`

**density ratio**.

**oblique shock wave**. Enter the

**upstream Mach number**and

**wave angle**to begin.