Flow velocity

The flow velocity , also flow velocity or flow velocity , is the velocity in a flow , a directed movement of particles or continuous bodies ( fluids ).

A distinction is made between the flow velocities of the individual particles and the mean flow velocity over a line, area or volume element or time interval.

The flow speed of water is the average speed at which the water moves downstream and is on the order of 1 meter per second. In contrast, that of groundwater is millimeters per second or centimeters per day, ie a few orders of magnitude lower.

definition

The flow velocity is the change in location of the individual point (location) along its path . ${\ displaystyle {\ vec {x}} = (x, y, z)}$

{\ displaystyle \ omega \, {\ text {or}} \, v \, {\ text {or}} \, c = | {\ vec {v}} | = {\ bigl |} {\ dot {\ vec {x}}} {\ bigr |}}

{\ displaystyle {\ vec {v}} = (v_ {x}, v_ {y}, v_ {z})}

: Flow

the point is the time derivative in physical notation

The flow velocity vectors carry one time line over into the next.

Average flow velocities can be determined using a streamline , the flow cross-section or the flow rate ( volume flow element , mass flow ).

Flow velocity in the potential field

In the potential field , the flow velocity follows the equation of motion

{\ displaystyle {\ dot {\ vec {v}}} = - {\ vec {\ nabla}} \ Phi}

{\ displaystyle \ nabla}

: Nabla operator

{\ displaystyle \ Phi}

: Potential

The specific energy equation applies :

{\ displaystyle {\ frac {v ^ {2}} {2}} + \ Phi = {\ text {const.}}}

Flow velocity in the Newtonian fluid

The flow velocity in a field of a flow in a Newtonian fluid is calculated from the Navier-Stokes equations , in their general formulation:

{\ displaystyle \ rho {\ frac {\ partial \ mathbf {v}} {\ partial t}} + \ rho (\ mathbf {v} \ cdot \ nabla) \ mathbf {v} = - \ nabla p + \ eta \ Delta \ mathbf {v} + (\ lambda + \ eta) \ nabla (\ nabla \ cdot \ mathbf {v}) + \ mathbf {f}.}

{\ displaystyle \ rho}

: Density of the fluid

{\ displaystyle t}

: Time

{\ displaystyle p}

: Pressure

{\ displaystyle \ lambda}

: 1. Lamé constant

{\ displaystyle \ eta}

: 2. Lamé constant, dynamic viscosity

{\ displaystyle {\ frac {\ partial} {\ partial t}}}

: Partial derivative with respect to time

{\ displaystyle \ nabla}

: Nabla operator

For this basic equation of fluid mechanics , a system of non-linear partial differential equations of the 2nd order, there are numerous simplifications, special cases and numerical approaches.

Non-Newtonian fluids such as blood, glycerine or dough, which show a non-proportional, erratic flow behavior (see rheology ), behave differently .

Application formulas

Examples of formulas in more specific application areas are:

The Bernoulli equation for smooth liquids and gases:

{\ displaystyle h + {\ frac {\ omega ^ {2}} {2g}} + {\ frac {p} {\ gamma}} = {\ text {const}}.}

Bernoulli's general law for frictionless, eddy-free , stationary flows
${\ displaystyle {\ frac {\ omega ^ {2}} {2}} + {\ frac {p} {\ rho}} = {\ text {const}}.}$

The Bernoulli law at constant density

{\ displaystyle \ rho {\ frac {v ^ {2}} {2}} + p = {\ text {const.}}}

Bernoulli equation with acceleration term for non-stationary flow:

{\ displaystyle h + {\ frac {\ omega ^ {2}} {2g}} + {\ frac {p} {\ gamma}} + {\ frac {1} {g}} \ int {\ frac {\ partial \ omega} {\ partial t}} = {\ text {const}}.}

Bernoulli equation of relative motion

{\ displaystyle {\ frac {\ omega ^ {2}} {2g}} + {\ frac {p} {\ gamma}} - {\ frac {u ^ {2}} {2g}} = {\ text { const}}.}

{\ displaystyle u}

: Circumferential speed - application for blade rings

Bernoulli equation at higher speeds (about> 150 m / s):

{\ displaystyle {\ frac {\ omega ^ {2}} {2g}} + \ int {\ frac {\ mathrm {d} p} {\ gamma}} = {\ text {const}}.}

Typical formulas for measurement methods

Dynamic pressure in the open air flow:

${\ displaystyle \ omega = {\ sqrt {\ frac {2q} {\ gamma}}}}$

{\ displaystyle q}

: Back pressure

Lateral discharge from an open vessel:

${\ displaystyle \ omega = {\ sqrt {2 {\ frac {\ Delta p} {\ gamma}}}}}$

Medium speeds

mean flow velocity with a non-constant cross-section
${\ displaystyle {\ bar {\ omega}} = {\ frac {1} {A}} \ cdot \ int _ {A} \ omega ({\ vec {x}}) \ cdot \ mathrm {d} A}$
Velocity at a point on the cross-section as a function of the location , with the direction of flow , so that . ${\ displaystyle f (x, y)}$ ${\ displaystyle z}$ ${\ displaystyle A = A (z)}$