THE CONSERVATION LAWS – mass conservation

1. THE CONSERVATION LAWS – mass conservation In the moment t the material volume V(t) contains a given amount of fluid that is described by mass m. The continuity condition states that there is no “free” space in volume V(t). The law of mass conservation states that the fluid in material volume V(t) within closed material surface A(t) can’t disappear or appear without the existence of internal sources or sinks. Integral form of massconservation law is the so-called continuity equation: Generally, the density within the control volume can be variable in space and time  =(t, r).

2. THE CONSERVATION LAWS – mass conservation After replacing the material volume V(t) with control volume V accompanying control surface A can be divided into "free" part A and part in contact with a rigid body S. Applying the transport theorem on transport property J = m and its density  = dJ/dV = dm/dV = one gets:

3. THE CONSERVATION LAWS – mass conservation The total change of mass in time consists of two parts. The first (on the right side of equation) is local and if one deals with non-steady flows it is different from 0. The remaining parts on the right hand side of equation are convective members representing the mass flux through the control surfaces A and S. Only the normal component of fluid velocity vnwill contribute to the amount of transported fluid, while tangential component of velocity vt influence only the deformations of control volume.

4. THE CONSERVATION LAWS – mass conservation If the velocity vector has the same orientation as outer normal on control surfaces dA and dS, the products and arepositive values. The other normal for control surface dS (around the body) is always pointed perpendicularly and toward to the solid body. Outflow from the “left” part of surface A' is the same as the inflow in the “right” part of surface A’.

5. THE CONSERVATION LAWS – mass conservation Using the GGO theorem: For transport equation (conservation of mass): one gets: The result of integration can be 0 only if the integral function is equal to 0. That gives rise to writing the differential form of continuity equation (conservation of mass): (valid for compressible as well as for incompressible fluids).

6. THE CONSERVATION LAWS – mass conservation For steady flows: local component /t = 0. For homogenous fluids: Since the fluid has finite density 0, follows Finally, continuity equation (conservation of mass) for 3D steady flow of incompressible fluid reads: For compressible fluids and unsteady flow with velocities less then speed of sound (water hammer) the continuity equation reads:

7. THE CONSERVATION LAWS – momentum conservation We are dealing with the fluid flow under the influence of forces (stresses). One derives the general equation that relates acceleration, mass forces and stresses on differential fluid element. The fluid is observed as continuum. Invoking the constitutive equations to define the relationships between stresses and strain velocity, one gets the so-called Navier-Stokes equation (NS). The base for the derivation of NS equation is the Newton second axiom: We are looking for the balance of external forces acting on the fluid element.

8. THE CONSERVATION LAWS – momentum conservation The total (absolute-substantial) change of momentum in time is defined with equation: Mass forces (e.g.. gravity acceleration g) can be quantified with potential gradient (e.g. gravitational potential =gz). Surface forces act on contact surfaces and can be decomposed on normal and shear components.

9. THE CONSERVATION LAWS – momentum conservation The force action is resolved due to action of surface stresses. In the Cartesian system we have an symmetric stress tensor with 3 diagonal members that represent normal stresses and 6 off-diagonal members that represent tangential stresses.

10. THE CONSERVATION LAWS – momentum conservation In 2D problems stress tensor has the shape: Writing the Taylor series with the inclusion of only the first member one holds (for x-direction):

11. THE CONSERVATION LAWS – momentum conservation Substitution in Newton second law-axiom (conservation of momentum) for x-direction holds : or in vector form for 3D problem: or by components :

12. THE CONSERVATION LAWS – momentum conservation The main problem is that we have 9 unknown components (3 velocity components and 6 stress components) and only 4 equations (1 continuity equation and 3 momentum equations). Constitutive equations define the stresses as function of velocities (more precisely velocity of deformation). Stress tensor consist of two parts. One is linked to the pressure p and other to the viscous stresses .

13. THE CONSERVATION LAWS – momentum conservation Pressure p represents isotropy normal stress independent on velocity or viscosity. Viscous stresses  in Newtonian fluids are linearly proportional to strain velocity (elements ei,j in symmetrical part of velocity gradient tensor) and viscosity : Using the dynamic viscosity coefficient as proportionality coefficient, tensor of viscous stresses in compact form reads:

14. THE CONSERVATION LAWS – momentum conservation For the Newtonian incompressible fluids the members of “total” stress tensor are given as: For example: Finally, introduction of constitutive equations in the second Newton axiom (momentum conservation) gives the so-called Navier-Stokes equation:

15. THE CONSERVATION LAWS – momentum conservation Now we have “closed” system of differential equations that is suitable for description of incompressible Newtonian fluid flow (4 equations and 4 unknowns - 3 velocity components and pressure):

16. THE CONSERVATION LAWS – Reynolds (RANS) Turbulent flow prevails in most engineering problems. According to the flow velocity measurements in one fixed point of the circular pressurized pipe we differentiate : 1 - statistically steady turbulent flow under statistically steady pressure condition, 2 - statistically unsteady turbulent flow under statistically unsteady pressure condition.

17. THE CONSERVATION LAWS – Reynolds (RANS) The instantaneous value of an arbitrary flow field (velocity, pressure, temperature) can be interpreted as the sum of average value (averaged over finite time interval – denoted with overbar) and fluctuating value (denoted with ‘ ): The averaged value of fluctuating component through the averaging period is equal to 0. The duration of averaging interval depends on the flow condition under consideration. This approach to the description of turbulence is stochastic in nature since the flow is viewed as a stochastic process with a random variable E. In following we comment only the turbulent flow of homogenous fluid.

18. THE CONSERVATION LAWS – Reynolds (RANS) Continuity equation for turbulent flow of homogenous and incompressible fluid has the same form regardless of the velocity form (instantaneous, averaged or fluctuating). After a few steps of algebraic manipulation and not taking into account the members of the higher order we get the so-called Reynolds equation: (component in x-direction)

19. THE CONSERVATION LAWS – Reynolds (RANS) Reynolds equation is derived from Navier-Stokes equation due to replacement of instantaneous velocities and pressure with the sum of averaged and fluctuating components. Reynolds equation and Navier-Stokes equation are similar. On the left hand side of Reynolds equation appear new members that represent the participation of fluctuation components on total momentum change in time. After replacing those members on the right hand side of equation, on the left hand side remain only the members that represent the change of averaged momentum in time.

20. THE CONSERVATION LAWS – Reynolds (RANS) On the right hand side in addition to the «real» averaged volume forces (gravitation) and averaged surface forces (pressure and viscosity) we also find additional «negative virtual» surface force associated with «virtual» stresses.

21. FLUID FLOW NEAR THE WALL - steady laminar flow The real fluid flow around the solid body or along its boundaries induces the force action on it. Those forces represent the resistance to the fluid flow and can be divided into two major components: Friction resistance due to tangential stresses (act on the surface of fluid and solid body), Form resistancedue to normal stresses (act normaly on the surfaces). Let’s analyze the situation of flow between two infinitely wide plates that are separated by relatively small distance B. In the first example upper plate is moving with constant velocity VB. The pressure gradient in movement direction is dp/ds = 0 (so-called Couette flow).

22. FLUID FLOW NEAR THE WALL - steady laminar flow The distribution of tangential stresses and velocities can be derived from Navier –Stokes equation for uniform and steady flow as well as from momentum conservation directly applied on control volume: According to adopted equality –dp/ds = 0 (Couette flow) we find the valid relation for the stress distribution along y-axes d/dy = 0.

23. FLUID FLOW NEAR THE WALL - steady laminar flow Observing the Newtonian fluid and laminar flow, the tangential stresses are defined through the linear relationship. Boundary conditions at contact surfaces between plates and fluid are: y = 0 u = 0 ; y = B u = VB. After integration one gets the linear profile for velocity distribution and constant for tangential stress distribution: In the next example both of the plates are in rest and at the angle to the horizontal plane. Pressure gradient in direction of flow is different from zero –dp/ds 0 (Poiseuille flow).

24. FLUID FLOW NEAR THE WALL - steady laminar flow Applying the momentum conservation principle on control volume reads: From geometrical relation sin = - dz/ds follows:

25. FLUID FLOW NEAR THE WALL - steady laminar flow The right hand side of equation represents the gradient of piezometer head with the symbol GP. After the use of Newton-constitutive equation  = (du/dy) we get: GP = -gIP = (d2u/dy2) Boundary conditions at contact surfaces between plates and fluid are y = 0 ; y = B u = 0. Parabolic velocity profile and linear tangential stress profile are obtained after integration: Maximum velocity is at coordinate (y = B/2) and has the value umax = -GPB2/8 . Mean velocity is V=2/3*umax.

26. FLUID FLOW NEAR THE WALL - boundary layer The flow within the boundary layer is non-uniform and develops in flow direction. Boundary layer covers the flow region between solid boundary and free flow. Flow within the boundary layer exhibits the properties of real – viscous fluid, while the free flow (beyond the boundary layer) can be observed as inviscid (ideal fluid).

27. FLUID FLOW NEAR THE WALL - boundary layer Let us analyze the example with the flat and thin plate in rest that is laid in a horizontal direction. The flow field is defined with uniform incoming velocity U0 until the plate is reached. One would note the development of boundary layer as increase in layer thickness between the plate and flow region with still undisturbed velocity U0 (free flow). In the boundary layer the velocity is a function of vertical distance from the plate u(y). The boundary layer thickness  increases along the plate, wherein the thickness is defined as the position where velocity reach the value u()=0,99U0.

28. FLUID FLOW NEAR THE WALL - boundary layer

29. FLUID FLOW NEAR THE WALL - boundary layer In the boundary layer may occur laminar and turbulent flow. At the beginning of the plate appears laminar boundary layer. After the initiation of instabilities the flow within the boundary layer turns from laminar to turbulent. Transition from laminar to turbulent flow occurs at the finite spatial distance (transition region). Local Reynolds number (no dimensional parameter) is used for parametisation Rex = U0x /, where x represents the distance from the plate beginning. Pressure gradient in the free flow is adopted as dp/dx = 0. Boundary layer is thin  the pressure in BL is constant. Boundary layer thickness depends on  =(x, U0, , ) = (t,). From the condition of dimensional homogenity:

30. FLUID FLOW NEAR THE WALL - boundary layer Non dimensional velocity profile in laminar boundary layer (for Rex< 500 000) after Blasius (1905): Tangential stress on the contact with the plate is derived from the velocity gradient directly:

31. FLUID FLOW NEAR THE WALL - boundary layer The turbulent boundary layer appears after the transition, and is characterized by much more complex structure of flow. Most of the boundary layer is made of a turbulent zone with eddies and fluctuations of flow parameters. In the vicinity of the wall the fluctuations are damped and laminar flow conditions prevail (viscous sublayer).

32. FLUID FLOW NEAR THE WALL - boundary layer Constant tangential stress 0 appears in the most part of BL. In the vicinity of free flow tangential stress decreases rapidly towards 0. One introduce the term „shear velocity“ : In the region of viscous sub-layer the pure laminar flow with linear velocity profile u = (0/) y takes place. In turbulentregion of BL the momentum exchange takes place mostly due to eddy activity (fluctuation in velocity field).

33. FLUID FLOW NEAR THE WALL - boundary layer As the result of more detailed analyses one gets the so-called logarithmic law of velocity distribution:  - Karman constant = 0,4 C - constant of integration as a function of boundary condition The value of constant C differs for the characteristic regions inside the overall turbulent region of boundary layer: For inner turbulent zone (strong viscosity influence): 30 < y / l < 500

34. FLUID FLOW NEAR THE WALL - boundary layer The transition from linear (viscous sublayer) to logarithmic law takes place in transition region 5<y/l<30 For other turbulent zone (negligible viscosity influence): y / l > 500 In practical application one can use simpler form for velocity distribution profile that is valid for a wide range of Re numbers: 85% of boundary layer is contained within other turbulent zone

35. FLUID FLOW NEAR THE WALL - boundary layer

36. BERNOULLI EQUATION - ideal fluid It is a useful tool in solving some engineering problems where pressure and mean velocities are to be determined. It deals with mechanical energy balance, following the changes in contribution of position, pressure and kinetic energies as the components of mechanical energy. Using the Bernoulli equation for an “ideal” fluid flow assumes that the fluid is inviscid and the flow is irrotational and steady. By adopting the idea of “inviscid” fluid, Navier-Stokes equation transforms to Euler equation :

37. BERNOULLI EQUATION - ideal fluid In the derivation of Bernoulli equation we use the Euler equation (component in z direction). Total change of velocity component w in z direction is given with (steady flow w/t=0) =0

38. BERNOULLI EQUATION - ideal fluid All members containing  vanish in irrotational flow: Bernoulli function is constant for z direction, along the streamline and perpendicular to it:

39. BERNOULLI EQUATION - ideal fluid Bernoulli equation enables monitoring and comparison of different states of flow along the streamlines, as well as the estimation of energy components contribution in the mechanical energy E as a whole: z (position energy level) + p/g (pressure energy level) = PL (piezometric level or piezometric head) PL (piezometric level) + v2/2g (kinetic energy level) = EL (mechanical energy level)

40. PRESSURIZED PIPE FLOW – circular cross section There is no free surface (incompressible Newton fluid). In this course we analyze only steady flow condition . The geometry of conduit (pipe) is defined with diameter D and length L. Applying the assumption L>>D we can ignore the influence of “initial” pipe section where velocity distribution develops up to the final stage. Downstream, the velocity profile is uniform if the pipe profile is also uniform. Stresses (friction) at the contact with the pipe are defined with Already known relation:

41. PRESSURIZED PIPE FLOW – circular cross section To preserve the flow, stress W need to be “overcome”. We want to determine the stress distribution in pipe cross section along with the empirical relationship between these stresses and velocity. The stress is linearly distributed perpendicular to the pipe axis:

42. PRESSURIZED PIPE FLOW – circular cross section If the flow is uniform, stress W has zero change in flow direction (du/ds=0; constant pipe cross section). Therefore, gradients of PL and EL are constant. The level of energy line on position 2 is less for ELIN (line loss of mechanical energy) in relation to position 1, regardless on the position of EL or PL (above or below pipe axis).

43. PRESSURIZED PIPE FLOW – circular cross section If PL lies below the pipe axis for some pipe section, the pressure is less then atmospheric. The occurrence of p<patmdoes not affect the slope of EL and PL and discharge through the pipe. Modification of the pipe length or diameter will cause the change in discharge (at the same boundary conditions - water level in reservoirs).

44. PRESSURIZED PIPE FLOW – circular cross section The intensity of line loss ELINcan be “easily” determined for the circular cross-section pipe with diameter D and length L. One would use the so-called Darcy-Weisbach friction coefficient  with the corresponding equation: The relationship between friction coefficient and stresses at the solid boundary wis defined with equation: Generally, the coefficient is the function of Re (Reynolds number) and /D (relative roughness). In the case of laminar flow:  = f (Re)= 64/Re (for Re<2300)

45. PRESSURIZED PIPE FLOW – circular cross section Analytical solution of Navier-Stokes equation does not exist for the flow in the turbulent regime (Re>2300). Consequently,  should be determined from experiment (Moody diagram). In the transitional regime is the function of Re number and relative roughness  = f (Re, /D), while in the turbulent-rough regime depends only on relative roughnes  = f (/D). In addition to line losses ELIN appear the local losses of mechanical energy ELOK. Its onset is induced by changing the pipe geometry and accompanied flow conditions (diameter expansion/narrowing, branches, closure vents). As in the case of line losses, local losses ELOK are calculated in relation to the kinetic energy (member v2/2g).

46. PRESSURIZED PIPE FLOW – friction resistance (line losses) For practical problems one can use explicit relation:

47. PRESSURIZED PIPE FLOW – friction resistance (line losses) For practical problems one can use explicit relation:

48. PRESSURIZED PIPE FLOW – friction resistance (local losses) A few examples of local losses :

49. PRESSURIZED PIPE FLOW – line and local losses

50. PRESSURIZED PIPE FLOW – pumps and turbines The pumps and turbines present the sources and sinks of mechanical energy. At the position of their occurrence appear local jump (pump) or local fall (turbine) in energy line (EL). The basic parameters in pump calculation are energy raising height HP and discharge QP that should be sustained in the pipe system.