Unsteady Flow

When using Bernoulli equation one of the assumptions that needs to be made is that the flow is steady.  For a flow that is steady, the velocity is only dependent on the distance s along the streamline.  In other words $v=v(s)$.  However if there is unsteady flow than time will have to also be considered in the function.  As a result $v = v(s,t)$.  The acceleration equation along the streamline for unsteady flow can be found using the following equation.

(Eq 1)  $a_s = \frac{∂v}{∂t}+v\frac{∂v}{∂s}$

If the flow were steady instead of unsteady than the time derivative would be removed from the above equation resulting in the following acceleration equation along a streamline.

(Eq 2)  $a_s =v\frac{∂v}{∂s}$

Unsteady Flow and the Bernoulli Equation

The above discusses how a particle will behave along a streamline if there is unsteady flow or steady flow.  However, this needs to be applied to Bernoulli equation.  In order to accomplish this you must realize that Bernoulli equation is obtained by integrating Newton’s Second Law of motion $F=ma$.  As a result the general form of Bernoulli equation will become the following.

(Eq 3) $p_1 + \frac{1}{2}ρv^2_1 + γz_1 = p_2 + \frac{1}{2}ρv^2_2 + γz_2$

$p$ = thermodynamic pressure

$ρ$ = fluid density

$v$ = fluid velocity

$γ$ = specific weight

$z$ = fluid height

In order to use the above equation the following assumptions will need to be satisfied.  The fluid is incomprehensible, the flow is steady, and the fluid is inviscid.  If all of these assumptions can be met than equation 3 can be used.  However, in this case we are saying that there is unsteady flow.  When this is the case the above equation must be modified.  If it is not modified than inaccuracies will result in the answer that you obtain.  In order to change the above equation you have remember that I stated that the general Bernoulli equation is obtained through integrating Newton Second Law.  As a result we will need to integrate the time derivative seen in equation 1 in respect to position 1 to position 2 along the streamline. As a result equation 3 will become the following.

(Eq 4) $p_1 + \frac{1}{2}ρv^2_1 + γz_1 =ρ\int{_{s_1}^{s_2}}\frac{∂v}{∂t}ds+ p_2 + \frac{1}{2}ρv^2_2 + γz_2$

Recall the above equation can only be used for unsteady flow along a streamline when these assumptions are satisfied. The fluid is incompressible and inviscid.  If those two assumptions cannot be satisfied than additional modifications to Bernoulli equation will need to be made.

 

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