Dear Nazaruddin Sinaga,

Basically the Bernoulli equation is an integral of the momentum equation for a frictionless flow. The exist many different versions of the Bernoulli equation. The most well known is the one that applies to steady, incompressible and frictionless flows p+rho U2= constant where p is the local "static" pressure, the the density (assumed constant) and U the magnitude of the local flow velocity . The equation of Bernoulli applies between 2 points (respectively A and B) along a streamline. When the flow originates from reservoir of uniform conditions (U=0, rho and p uniform), the same constant p+U2/2=Po applies in the entire flow (hence allow between points not located on a single streamline).

The compressible version is h+U2/2=constant, where h is the enthalpy. For an isentropic flow (no friction, no heat transfer) the enthalpy is the integral of dp/rho. This equation is very useful in Gas Dynamics (the study of compressible flows: Compressible Fluid Dynamics (Advanced engineering series) by Philip A. Thompson (1972-02-03) and Landau en Lifchitz theoretical Physics Vol. VI http://users-phys.au.dk/srf/hydro/Landau+Lifschitz.pdf).

The Bernoulli equation for an unsteady flow is dPhi/dt +h+ U2/2=constant between point A and B along a streamline. Here dPhi/dt is the PARTIAL time derivative of the potential Phi defined such that grad Phi is the local velocity vector vex(v) (note |vec(v)|=U). See the book of Patterson (Paterson, A. R., A First Course in Fluid Dynamics. Cambridge et al., Cambridge University Press 1983).

In linear acoustics this equation becomes

dPhi/dt+p'=0, where p' is the acoustic perturbation and U is assumed to be small compared to the speed of sound c so that U2/c2>1). The Strouhal number Sr=L f/U (where f is a characteristic frequency of the "unsteadiness") should be small (Sr

You already wrote the hypothesis for apply the Bernoulli integral, it can be extended to unsteady case but it implies that the total pressure is invariant which is far from being true in a pipe flow. Correction is required.

Thanks Filippo Maria Denaro

Yes, basically, it applies only for steady flow...

Nazaruddin Sinaga

you can extend also to unsteady flow but the real problem for the validity of the Bernoulli integral is the hypothesis that the flow has no dissipative effects.

A lesser strict set of hypotheses can be found in the Crocco relation.

First of all what do you mean with actual situation? the difference will be significant because of fluid losses .However Bernoulli equation is not enough to specify of explore the real situation.

Dear Nazaruddin Sinaga,

Basically the Bernoulli equation is an integral of the momentum equation for a frictionless flow. The exist many different versions of the Bernoulli equation. The most well known is the one that applies to steady, incompressible and frictionless flows p+rho U2= constant where p is the local "static" pressure, the the density (assumed constant) and U the magnitude of the local flow velocity . The equation of Bernoulli applies between 2 points (respectively A and B) along a streamline. When the flow originates from reservoir of uniform conditions (U=0, rho and p uniform), the same constant p+U2/2=Po applies in the entire flow (hence allow between points not located on a single streamline).

The compressible version is h+U2/2=constant, where h is the enthalpy. For an isentropic flow (no friction, no heat transfer) the enthalpy is the integral of dp/rho. This equation is very useful in Gas Dynamics (the study of compressible flows: Compressible Fluid Dynamics (Advanced engineering series) by Philip A. Thompson (1972-02-03) and Landau en Lifchitz theoretical Physics Vol. VI http://users-phys.au.dk/srf/hydro/Landau+Lifschitz.pdf).

The Bernoulli equation for an unsteady flow is dPhi/dt +h+ U2/2=constant between point A and B along a streamline. Here dPhi/dt is the PARTIAL time derivative of the potential Phi defined such that grad Phi is the local velocity vector vex(v) (note |vec(v)|=U). See the book of Patterson (Paterson, A. R., A First Course in Fluid Dynamics. Cambridge et al., Cambridge University Press 1983).

In linear acoustics this equation becomes

dPhi/dt+p'=0, where p' is the acoustic perturbation and U is assumed to be small compared to the speed of sound c so that U2/c2>1). The Strouhal number Sr=L f/U (where f is a characteristic frequency of the "unsteadiness") should be small (Sr

Dear Avraham Hirschberg

Thank you very much for your complete explanation. It is very interesting and some are new for me. So you say that:

1) The choice of characteristic length scale L and frequency f, as critical for a good result.

2} Turbulent pipe flows are essentially unsteady, and determining a streamline to apply Bernoulli is a very difficult to determine.

3) in a steady flow in a long pipe, the convective acceleration vanishes.

3) Re=U D/nu is not a measure for the importance of friction.

Please correct me if I am wrong.

Thank you

Nazar

Globally yes:

1) Yes

2) Frictional dissipation in turbulent flow is very efficient (energy cascade to dissipative scales), hence turbulent flows are only very locally frictionless.

3) Indeed, in a very long pipe with fully developed velocity profile, the convective acceleration in the main flow direction vanishes. The Reynolds number yields an indication for transition to turbulence not for friction.

4)Re=U D/nu is in a long pipe not a measure for friction. A much better scale for friction is (D/L)sort(U D/nu) where D is the pipe diameter and L the pipe length.

Regards,

Mico

Thanks Avraham Hirschberg

May I have a question, please?

Nazaruddin Sinaga

It is very instructive, from an educational point of view, to evaluate quantitative the error in the Bernoulli integral for the simple case of a real laminar viscous flow in a channel. Select a viscosity, a flow rate, the distance between the walls to express the pressure decay over a lenght L in the Poiseulle solution. Since the velocity is constant along the streamline, you can simply evaluate the decay of the total pressure. Do not forget that Bernoulli is nothing else that an energetic integral.

Thanks Filippo Maria Denaro

I appreciate your reply. Can we derive the Bernoulli equation not from the streamline coordinate, so we can neglect the assumption that the equation must be in the streamline?

Thank you....

Nazaruddin Sinaga

when you derive Bernoulli, you actually get a vector relation in which the gradient of the total pressure vanishes. You can project such vector equation along a certain direction r getting that the directional derivative d/dr=r.Grad(total pressure) vanishes.