Consider a real fluid flowing past a cylinder of length L and diameter D. the drag coefficient is,

`F_D=C_D*(rho*U^2)/2*A`

Where A is the projected area, which is equal to `LD` for a cylinder.

At very small velocities, `R_e<0.5` the fluid sticks to the cylinder all the way round and never separates from the cylinder. This produces a streamline pattern similar to that of an ideal fluid as shown in figure. The dynamic pressure is achieved at the front stagnation point and vacuum equal to the dynamic pressure exists at the top and bottom where the velocity is at its greatest. 

As the velocity increases, the boundary layer breaks away and eddies are formed behind as shown in figure. The drag pressure increases at the front due to the pressure behind and drag pressure drops at the back. 

A further increase in velocity cause eddies to elongate and the drag coefficient becomes nearly constant as shown in figure. The pressure distribution shows that ambient pressure exists at the rear of the cylinder.

At Reynolds number of around 90, the vortices break away alternatively from the top and bottom of the cylinder producing a vortex street in the wake as shown in figure. The pressure distribution shows the vacuum at the rear.

Drag Coefficient as a function of Reynolds Number:

• For `R_e<1`the flow pattern will be symmetrical.

• For `R_e =1 1 to 2000, `C_D` decreases and attains a minimum value of 0.95 at `R_e`=2000.

• For `R_e=2000` to `3*10^4`, `C_D` increases and become 1.2 at `R_e=3*10^4`

• For `R_e=3*10^4` to `3*10^5`, `C_D` decreases and its value becomes 0.3 at a value of `R_e=3*10^5`

• For `R_e=3*10^6`, `C_D` increases and attains a value of o.7 in the end.

Consider a real fluid flowing past a cylinder of length L and diameter D. the drag coefficient is,

`F_D=C_D*(rho*U^2)/2*A`

Where A is the projected area, which is equal to `LD` for a cylinder.

At very small velocities, `R_e<0.5` the fluid sticks to the cylinder all the way round and never separates from the cylinder. This produces a streamline pattern similar to that of an ideal fluid as shown in figure. The dynamic pressure is achieved at the front stagnation point and vacuum equal to the dynamic pressure exists at the top and bottom where the velocity is at its greatest. 

As the velocity increases, the boundary layer breaks away and eddies are formed behind as shown in figure. The drag pressure increases at the front due to the pressure behind and drag pressure drops at the back. 

A further increase in velocity cause eddies to elongate and the drag coefficient becomes nearly constant as shown in figure. The pressure distribution shows that ambient pressure exists at the rear of the cylinder.

At Reynolds number of around 90, the vortices break away alternatively from the top and bottom of the cylinder producing a vortex street in the wake as shown in figure. The pressure distribution shows the vacuum at the rear.

Drag Coefficient as a function of Reynolds Number:

• For `R_e<1`the flow pattern will be symmetrical.

• For `R_e =1 1 to 2000, `C_D` decreases and attains a minimum value of 0.95 at `R_e`=2000.

• For `R_e=2000` to `3*10^4`, `C_D` increases and become 1.2 at `R_e=3*10^4`

• For `R_e=3*10^4` to `3*10^5`, `C_D` decreases and its value becomes 0.3 at a value of `R_e=3*10^5`

• For `R_e=3*10^6`, `C_D` increases and attains a value of o.7 in the end.

Streamlined Body:

A streamlined body is a body having a shape that lowers the friction drag between a fluid, like air and water, and an object moving through that fluid. In other words, A streamlined body is defined as that body whose surface is aligned with the streamlines, when the body is placed in the flow.

 Streamlined body experiences drag mainly due to Viscous/Frictional drag, which depends on the surface area exposed to fluid. Since the drag on the streamlined body is low, the lift/drag ratio is high.In this case, the separation usually takes place at the trailing edge.

An example of streamlined body is a thin aerofoil.

Bluff Body:

 A bluff body is defined as that body whose surface is not aligned with the stream-lines, when placed in the flow. Thus, the body offers less resistance in terms of Viscous/Frictional drag. There is a very large pressure drag, due to eddy formation after the body leading to a large wake region. Thus the pressure drag depends on the cross sectional area of the body rather than the surface area exposed.

Streamlined Body:

A streamlined body is a body having a shape that lowers the friction drag between a fluid, like air and water, and an object moving through that fluid. In other words, A streamlined body is defined as that body whose surface is aligned with the streamlines, when the body is placed in the flow.

 Streamlined body experiences drag mainly due to Viscous/Frictional drag, which depends on the surface area exposed to fluid. Since the drag on the streamlined body is low, the lift/drag ratio is high.In this case, the separation usually takes place at the trailing edge.

An example of streamlined body is a thin aerofoil.

Bluff Body:

 A bluff body is defined as that body whose surface is not aligned with the stream-lines, when placed in the flow. Thus, the body offers less resistance in terms of Viscous/Frictional drag. There is a very large pressure drag, due to eddy formation after the body leading to a large wake region. Thus the pressure drag depends on the cross sectional area of the body rather than the surface area exposed.

Let us consider a case when real fluid flows past a sphere. Let D be the diameter of the sphere, v be the velocity of flow pf fluid of mass density `rho` and viscosity `mu`.

We have,

`R_e=(rho*U*D)/mu`

When velocity of flow is very small or the fluid is very viscous such that the Reynolds number is very small (`R_e<=0.2`), then the viscous forces are more predominant than the inertial force. Stokes analysed theoretically the flow around sphere under `R_e<0.2` and found that the total drag force is,

`F_D=3pimuDU`....(i)

He also found that out of total drag, two third is contributed by skin friction and one third by pressure difference.

Thus,

Skin friction drag`=2/3F_D=2pimuDU`

Pressure drag `=1/2F_d=pimuDU`

Again, the total drag is,

`F_D=C_D*(rho*U^2)/2*A`

Where `A` is the projected area of sphere.

`A=(pi*D^2)/4`

Thus,

`C_D*(rho*U^2)/2*A=3pimuDU`

`C_D=(24mu)/(rhoUD)=24/R_e`

• For `R_e` between 0.2 and 5

• `C_D=24/R_e(1+3/(16R_e))` this is called onseen formula.

• For `R_e` between 5 and 1000, `C_D=0.4`

• For `R_e` between 1000 and 100,000, `C_D=0.5`.

• For `R_e>10^5`, `C_D=0.2`.

Let us consider a case when real fluid flows past a sphere. Let D be the diameter of the sphere, v be the velocity of flow pf fluid of mass density `rho` and viscosity `mu`.

We have,

`R_e=(rho*U*D)/mu`

When velocity of flow is very small or the fluid is very viscous such that the Reynolds number is very small (`R_e<=0.2`), then the viscous forces are more predominant than the inertial force. Stokes analysed theoretically the flow around sphere under `R_e<0.2` and found that the total drag force is,

`F_D=3pimuDU`....(i)

He also found that out of total drag, two third is contributed by skin friction and one third by pressure difference.

Thus,

Skin friction drag`=2/3F_D=2pimuDU`

Pressure drag `=1/2F_d=pimuDU`

Again, the total drag is,

`F_D=C_D*(rho*U^2)/2*A`

Where `A` is the projected area of sphere.

`A=(pi*D^2)/4`

Thus,

`C_D*(rho*U^2)/2*A=3pimuDU`

`C_D=(24mu)/(rhoUD)=24/R_e`

• For `R_e` between 0.2 and 5

• `C_D=24/R_e(1+3/(16R_e))` this is called onseen formula.

• For `R_e` between 5 and 1000, `C_D=0.4`

• For `R_e` between 1000 and 100,000, `C_D=0.5`.

• For `R_e>10^5`, `C_D=0.2`.