# Turbulent Flow: Reynolds Number, Shear Stress, Equation

By BYJU'S Exam Prep

Updated on: September 25th, 2023

**Turbulent Flow** is a type of fluid flow. It may occur in the open channel or in the pipes. Based on the motion of the fluid flow, it can be classified into the laminar flow, turbulent flow or the transition between these types. These types of flow are classified based on the Reynolds number of the fluid flow.

### Turbulent Flow PDF

Reynolds number of a flow depends on the flow and channel characteristics. If Reynold’s number of a flow in a pipe is greater than 2000, it is classified as a turbulent flow. In such a type of flow, a fluid particle does not flow in a layered form and is mixed with others. This article contains basic notes on the “Turbulent Flow” topic of the “Fluid Mechanics & Hydraulics” subject.

Table of content

## What is Turbulent Flow?

Turbulent flow is the type of flow that can occur in a closed pipe or in an open channel. Flow categorization depends on the Reynolds number of the flow. It can be laminar flow, turbulent flow or the transition between laminar and turbulent state. This type of flow is essential for the GATE CE exam. Turbulent flow is a flow regime characterized by the following points as given below.

Shear stress in the turbulent flow

Τ = τ_{v} +τ_{t} = μ (du/dy) + η (du/dy)

where

- τ
_{v}and τ_{t }= shear stress due to viscosity and turbulence. - η = eddy viscosity coefficient.

### Turbulent Shear Stress by Reynolds

τ = ρu’v’

u’ and v’ fluctuating components of velocity.

## Shear Stress in Turbulent Flow

Shear stress is the value of the force applied on the pipe wall per unit cross-section area. In the turbulent flow, the fluid particles will move in a random direction that will cause a shearing force on the pipe wall. Hence shearing stress will be generated on the wall. Shear flow in turbulent flow can be explained below.

Τ = ρl^{2} (du/dy)^{2}

where, l = Mixing length

The expression gives the velocity distribution in the turbulent flow for pipes.

u = u_{max} + 2.5 u* log_{e}(y/R)

U_{max} = center velocity

where,

- y = Distance from the pipe wall,
- R = radius of the pipe

u* = Shear velocity = (τ_{0}/ρ)^{0.5}

Velocity defect is the difference between the maximum velocity (u_{max}) and local velocity (u) at any point given by

u_{max} – u = 5.75 u*log_{10}(R/y)

## Turbulent Flow Reynolds Number

Reynolds number is a dimensionless number of the flow used to represent the flow characteristics. And it is defined as the ratio of the Inertial force to the viscous force. Mathematically, it can be defined as R_{e} = ρVL/μ.

Based on the value of the Reynolds number, the flow will be characterized as either laminar flow or turbulent flow. For the pipe flow, if the value of Reynold number is greater than 2000, it is classified as flow is turbulent. If the Reynolds number of the flow is less than 500, it is classified as laminar flow, and if the Reynolds number lies between 500 to 2000, it will classified as the transition flow.

## Karman Prandtl Velocity Distribution Equation in Turbulent Flow

The velocity distribution equation in turbulent flow follows logarithmic distributions, while in laminar flow, velocity distribution follows the parabolic distribution. Karman Prandtl gives a velocity distribution equation for the turbulent flow through a pipe, the hydrodynamically smooth pipe and rough pipes. It will explain further:

Hydrodynamically pipe

where

- u = velocity at any point in the turbulent flow
- u* = shear velocity = (τ
_{0}/ρ)^{0.5} - v = Kinematic viscosity of the fluid
- y = Distance from the wall of the pipe
- k = Roughness factor

Velocity distribution in terms of average velocity

## Common Mean Velocity Distribution Equation

Common mean velocity is the velocity of the fluid for which the same discharge and energy will be found as that of the actual flow. Actual velocity distribution for the turbulent flow will follow logerthic variation explained as below.

**[(u-u¯)/ u*]= 5.75 log _{10 }(y/R) + 3.75 **

(This is valid for both rough and smooth pipes, that’s why it is referred to as Common Mean Velocity Distribution Equation)

Coefficient of friction

f = 16/R_{e} (for laminar flow)

f = 0.0791/(Re)^{0.25} ; 4000≤R_{e}<10^{5}

f = 0.0008 + 0.05525/(R_{e})^{0.257} , 4×10^{7}≤R_{e}≤10^{5}

(for smooth pipe)

1/(4f)^{0.25} = 2log_{10}(R/k) + 1.74 (for rough pipe)

## Difference Between Laminar Flow and Turbulent Flow

Laminar and turbulent flow are both conditions of a fluid flow. These conditions depend on the Reynolds number of the flow and are very important for the GATE CE question paper. Here a few points explain the difference between laminar and turbulent flow.

- In the laminar flow, fluid particles move in a layered form, while in the turbulent flow, the fluid particle moves haphazardly.
- Velocity distribution in laminar flow follows parabolic distribution, while in turbulent flow, it varies in a logarithmic way.
- The magnitude and direction both of velocities vary for the turbulent flow but for the laminar flow, only the magnitude of the velocity varies.
- Conditions for Reynolds number for both types of flow are different, and it also depends on the type of conduit, whether it is open or closed.

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