**critical Reynolds number**is associated with the

**laminar-turbulent transition,**in which a laminar flow becomes turbulent.

For flow in a pipe of diameter *D*, experimental observations show that for “fully developed” flow, the critical Reynolds number is about **Re _{d,crit} = 2300.**

For flow over a flat plate, **transition from laminar to turbulent** boundary layer occurs when Reynolds number at x exceeds **Re _{x,crit}**

**~ 500,000**.

**The Reynolds number** is the ratio of **inertial forces **to **viscous forces** and is a convenient parameter for predicting if a flow condition will be **laminar or turbulent**.

The **critical Reynolds number** is associated with the **laminar-turbulent transition, **in which a laminar flow becomes turbulent. This is an extraordinarily complicated process, which is not fully understood at present.

The value of the critical Reynolds number is different for different geometries.

- For flow over a flat plate, the generally accepted value of the critical Reynolds number is
**Rex ~ 500000**. - For flow in a pipe of diameter D, experimental observations show that for “fully developed” flow, laminar flow occurs when ReD < 2300, and turbulent flow occurs when
**ReD > 3500**. - For a sphere in a fluid, the characteristic length-scale is the diameter of the sphere, and the characteristic velocity is that of the sphere relative to the fluid. Purely laminar flow only exists up to
**Re = 10**under this definition.

## Critical Reynolds Number for Flow in a Pipe

For flow in a pipe of diameter *D*, experimental observations show that for “fully developed” flow, the critical Reynolds number is about **Re _{d,crit} = 2300.**

**Laminar flow.**For practical purposes, the flow is laminar if the Reynolds number is**less than 2000**. The accepted transition Reynolds number for flow in a circular pipe is**Re**_{d,crit}= 2300.**Transitional flow.**At Reynolds numbers**between about 2000 and 4000,**the flow is unstable due to the onset of turbulence. These flows are sometimes referred to as transitional flows.**Turbulent flow.**If the Reynolds number is**greater than 3500**, the flow is turbulent.

Note that the critical Reynolds number is different for every geometry.

## Critical Reynolds Number in boundary layer flow over a flat plate.

In determining whether the boundary layer is laminar or turbulent, it is frequently reasonable to assume that transition begins at some location **x _{crit}**, as shown in the figure. This location is determined by the critical Reynolds number,

**Re**. For flow over a flat plate,

_{x,crit}**transition from laminar to turbulent** boundary layer occurs when Reynolds number at x exceeds **Re _{x,crit}**

**~ 500,000**. The transition may occur earlier, but it is dependent especially on the

**surface roughness**. The turbulent boundary layer thickens more rapidly than the laminar boundary layer due to increased shear stress at the body surface.

## Example: Critical Reynolds Number

A long thin flat plate is placed parallel to a **1 m/s** stream of water at** 20°C**. Assume that the kinematic viscosity of water at 20°C is equal to **1×10 ^{-6} m^{2}/s**.

At **what distance x** from the leading edge will be the **transition** from laminar to turbulent boundary layer (i.e., find Re_{x} ~ 500,000).

**Solution:**

To locate the transition from laminar to the turbulent boundary layer, we have to find x at **Re _{x} ~ 500,000**.

**x** = 500 000 x 1×10^{-6} [m^{2}/s] / 1 [m/s] = **0.5 m**