Continuity and Navier-Stokes Equations Lecture 11 and 13 Mecânica de Fluidos Ambiental 2015/2016.

Mecânica de Fluidos Ambiental 2015/2016

Continuity and Navier-Stokes EquationsLecture 11 and 13


Previous lectures• In previous lectures attention is focused in the use of

finite control volume for the solution of a variety of fluid mechanics problems.

• This approach is pratical and useful, since it does not generally require a detail knowledge of pressure and velocity variation within the volume control.

• Unfortunately, there are many situations in which the details of the flow are important and the finite control volume approach not yield the desired information:• How the velocity varies over the cross section of a pipe• How the pressure and shear stress vary along the surface of an

airplain wing


This lectures• In these circunstances we need to develop relationships that apply

to a point, or at least a very small region (infinitessimal volume) within a given flow field.

• This approach is commonly referred to as differential anlysis, since the governing equations are differential equations.

• We will provide an introduction to the diferrential equation that describe (in detail) the motion of fluids.

• Unfortunately, we will also find that these equations are rather complicated, partial differential equations cannot be solved exactly except in a few cases, at least without making some simplifying assumptions.

• we will perform the derivation of the basic differential equations, which will be based on the principle of conservation of mass and Newton’s second law of motion


Mass conservation

• Mass is conserved • This implies that the rate of accumulation inside a

volume balances the input and output budgets:

• If fluid the is incompressible• Control volume infinitessimal

)(.. SiSodAncnudVt surfacevc


• =

tdxdydzdV

tdV

t vcvc

dzzzzz

dyyyyy

dxxxxxsurface

udxdyudxdy

udxdzudxdz

udydzudydzdAnu

.

dzzzzz

dyyyyy

dxxxxx

udxdyudxdy

udxdzudxdz

udydzudydzt

dxdydz


• Dividing all the equation by the volume

dzuu

dy

uu

dxuu

tdzzzzzdyyyyydxxxxx

zu

yu

xu

tzyx

i

i

i

i

ii

i

i

xu

dtd

xu

xu

t

xu

t

0

i

i

xu

This is minus the divergence that is the difference between what is flowing out minus what is flowing in. Einstein convection dvi/dxi= dv1/dx1+dv2/dx2+dv3/dx3. If divergence is positive it means that there is more volume getting out that getting in, the volume is increasing.

Fluid incompressible:

If we shrink the volume to a point:

Continuity equation


Interpretation

i

ixu

dtd

i

ixu

t

i

i

ii x

ux

ut

0

i

i

xu

The local derivative is proportional to the local mass flux divergence. Meaning that in a local infinitesimal volume the rate of change of density is proportional to the incoming flux minus the outgoing flux. If these fluxes balance (null divergence) there is no mass accumulation and consequently there is no change of local density

The total derivative is proportional to the local velocity divergence. As the fluid moves its density changes according to the local divergence. If it is negative there is contraction consequently the density increases.

If the flow is stationary the local derivatives are null. In this case the advection, i.e. the rate at which the property leaves minus the rate at which it enters balances the local production (i.e. the difference between what leaves and enters is the production)

If the fluid is incompressible there is no accumulation and the density entering and leaving are equal. As a consequence there is no production.


Another way of getting the equation• Based on the assumption that mass remains constant

• The Reynolds theorem states that:

0dtdm

0

dt

dVoldSystem

dSnvdVoldtddVol

t VC SCsistema

.

dSnvdVolt VC SC

.


Momentum conservation

• One must evaluate the forces per unit of volume:• Surface forces: shear and normal stress• Body (“volume”)forces: gravity

dtudmF

dtud

VolF

=>


Pressure force


Resultant force(including friction and weight)

dxdydzgWeight

dyyyx

yyx


Resultant force• Summing all the forces:

dxdydzgWeight

dyyyx

yyx

ixppressure

dxdydz

dxdydxdydz

dxdz

dxdydzdydz

Friction zzidzyziyyidyyyixxidxxxi

dxdydzgweight

ij

ji

ii g

xxp

dtdu

This is the Momentum transport equation into its differential form. It states that the fluid acceleration is the result of the 3 forces (pressure, friction and gravity).


Case of momentum equation• For incompressible Newtonian fluids it is known that the

stresses are linearly related to the rates of deformation and can be expressed:

i

j

ji

ijji xu

xu

ii

ii xu

2

Replacing in the momentum evolution equation:

ijijij

ij

ii gxx

pxuu

tu

dtdu

One gets:

iji

jiji

jii g

xu

xxp

xuu

tu

dtdu

This is the Navier-Stokes Equation, i.e., the Momentum evolution equation of a incompressible Newtonian fluid.


Navier-stokes equation

• They are second-order nonlinear partial differential equations and are quite formidable, but solutions have been found to a variety of interesting viscous flow problems

• Equations (4.38) have four unknowns: p, u, v and w. They should be combined with the incompressible continuity relation to form four equations in these four unknowns.


Viscosity• Friction force (shear) along a surface appears when molecules from

one side cross to the other and velocities on both sides are different.

• The gradient responsible for shear is the velocity gradient and not the gradient of momentum.

• Velocities being different on both sides of the surface require the velocity of the “migrating molecules” to change sharing momentum with the local molecules. As a consequence local velocity changes and momentum changes as well. The change of velocity generates a inertia force proportional to the density . Thus the diffusive flux of momentum is equal to the diffusive flux of velocity multiplied by the density:

su

su


Synthesis

• A tensão de corte é o fluxo difusivo de quantidade de movimento.

• A tensão de corte é tangente à velocidade e origina um fluxo de quantidade de movimento perpendicular à velocidade, no sentido contrário do gradiente de velocidade.

• A velocidade tem 3 componentes e por isso o seu gradiente tem 9 (cada uma das 3 componentes pode variar nas 3 direcções do espaço).


Hipóteses Subjacentes Tensão de corte em fluidos Newtonianos


• Pensemos num volume infinitesimal com a forma de um cubo e na componente “1” da velocidade (representada a verde).

• Esta componente pode varia na direcção “1”, na direcção “2” e na direcção “3”, dando origem respectivamente às tensões:

;31;21;11


• Estas tensões actuam nas faces do cubo, cujas normais não na direcção “1”, ou “2”, ou “3”. Existem duas faces para cada uma das direcções.

• A resultante das forças é a diferença entre as tensões que actuam em faces correspondentes, que por unidade de volume dá:

• A convenção de sinais é: o que entra é positivo

1jjx


Como determinar as tensões?

• São proporcionais ao gradiente de velocidade

• Não pode haver efeito de pressão(a força é tangencial):

• E o momento resultante sobre um volume de controlo tem que ser nulo (caso contrário teria aceleração angular).

0332211


Caso geral

dyyx

yx

xy

dxxy

yxxy


Expressão geral da Tensão de corte

i

j

j

iji x

uxu

Quando i=j esta expressão dá a divergência da velocidade, que se o fluido for compressível tem que ser anulada. A expressão geral fica:

k

kij

i

j

j

iji x

uxu

xu

32

Continuity and Navier-Stokes Equations Lecture 11 and 13 Mecânica de Fluidos Ambiental 2015/2016.

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Transcript of Continuity and Navier-Stokes Equations Lecture 11 and 13 Mecânica de Fluidos Ambiental 2015/2016.