6 Ch 7 - Dimensional analysis and similitude v1 PDF

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10/30/2018

Dimensional analysis and similitude Dimensional analysis refers to a method of identifying non-dimensional parameters associated with a physical phenomenon. Identifying the appropriate dimensionless parameters provides insight into the physics of the flow and potentially reduces the number of test runs needed to understand the relationships that may exist between various parameters and the underlying physics. In addition, consolidation and interpretation of experimental results can be greatly simplified. Similitude refers to the idea that we can scale various parameters in the identified non-dimensional groups and as long as the group magnitude stays the same, the scaling maintains that similar physics are occurring. - Non-dimensionalizing the governing equations (4) - Nature of dimensional analysis (3) - Buckingham Pi Theorem (5) - Similitude and model scaling (6)

Mark McQuilling

ESCI 3200 Fluid Dynamics

v 1.2

Saint Louis University

Slide 1

Non-dimensionalizing the governing equations (1) Non-dimensionalizing the governing equations can make them more generally applicable and can provide more insight into fluid behaviors. We’ll assume constant gravity and specific heats. First we need to select constant reference parameters appropriate to the flow: 1. reference velocity – usually U  for external flows and U e for internal flows 2. reference length, L , usually a body length for external flows or a duct diameter for internal flows 3. freestream properties P ,  , T ,  , k 4. characteristic time – steady viscous flows have no characteristic time, so typically the particle residence time L / U  is used for a reference time - if applicable, use oscillation frequency  , or shedding frequency f , etc. • Now we can define dimensionless variables, denoted by an asterisk: x*i 

xi L

* 

 

Mark McQuilling

tU t   L *

T *

T  T Tw  T

  V V * U

* 

ESCI 3200 Fluid Dynamics

 

P* 

P  P  U2

k*  v 1.2

k k

* 

L2  U 

 *  L Saint Louis University

Slide 2

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10/30/2018

Non-dimensionalizing the governing equations (2) Substitution of the variables into the governing equations produces a more general set of the equations. For example, let’s look at continuity: 

* L

  V  V *U

  * 

t *L U

t

     ( V )  0 t

• Now substitute:

*   *   U  * * * *  L   [(   )(V U  )]  L   ( V )  

 U  * (  * )    * *  (t L /U  ) L t

 U   * L

t *



U  L

 * ( *V * )  0

  *  *  (  * V * )  0 * t Mark McQuilling

ESCI 3200 Fluid Dynamics

v 1.2

Saint Louis University

Slide 3

Non-dimensionalizing the governing equations (3) The same procedure works for the momentum and energy equations:

*

 * DV * 1 * 1 *  *   u *i u j  * * P           *   * Dt * FR Re    x j x i 

*

DT * DP* 1 Ec *  Ec *   *  ( k** T * )   * Re Pr Re Dt Dt

• So we see the non-dimensional continuity equation matches the dimensional version, but the momentum and energy equations contain similarity parameters (dimensionless groups). Two flows in similar geometries are called physically similar (think flow physics) when all of their similarity parameters are equal. • Navier-Stokes equations contain two similarity parameters: 1. Reynolds number: Re  (inertial to viscous forces)

 U L 

Fr 

U 2 gL

Ec 

U 2 cp T

2. Froude number: (bulk fluid to surface wave speed)

• energy equation contains two more similarity parameters: 3. Prandtl number: (viscous to thermal diffusion rate) Mark McQuilling

Pr 

 c p k

ESCI 3200 Fluid Dynamics

4. Eckert number:

(directed energy to internal energy) v 1.2

Saint Louis University

Slide 4

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Non-dimensionalizing the governing equations (4) When working with non-dimensional equations, we also need non-dimensional boundary conditions, found by using the same reference quantities as before. V * 1 • In the freestream, where V  U and T  T , the boundary conditions become: T * 0 • At a solid, rigid wall: Vw*  0, Tw*  1 - Nusselt # compares 1st kind convection to conduction rates

or

 * T* qw L  k   Nu *     w k ( Tw  T ) 2nd kind

  u*  uw*  Kn  *  • At slip walls or those with a temperature jump:    w - Knudsen # compares molecular  T *  2 to domain length scales Tw*  1  Kn  *  Pr( 1)    w

• If we have a free surface, where 1   * 1  1 *   * surface tension is important, the Pinterface  Ca  *  Fr We  Rex Rey  interface pressure becomes:  U2 L Pa  P We    Ca   (Cavitation # = local pressure to momentum)  U2 (Weber # = momentum to surface stress) Mark McQuilling

ESCI 3200 Fluid Dynamics

v 1.2

Saint Louis University

Slide 5

Nature of dimensional analysis (1) The last section showed how non-dimensionalizing the governing equations and associated boundary conditions can provide insight into general fluid behaviors (say at low and high Reynolds numbers). This section of notes shows some of the results of plotting various dimensionless groups together – pay attention to the understanding that can be had without doing any (more) tests! • drag force non-dimensionalized by dynamic pressure and projected area (also called drag coefficient) is potted against Reynolds number for a sphere in cross-flow - any test on a sphere in any fluid will produce the same result if non-dim’d the same way • plotting these two non-dimensional groups together shows the relationship, but does not provide understanding of why the relationship exists (i.e. the underlying physics) - other groups plotted together may or may not show a trend - if there is a trend, the relationship is called a correlation (or, the variables are correlated), which means they are “co” – “related”, or related to each other Mark McQuilling

ESCI 3200 Fluid Dynamics

v 1.2

Saint Louis University

Slide 6

3

10/30/2018

Nature of dimensional analysis (2) • here’s a component performance map for the compressor section of a gas turbine engine, showing relationships between pressure coefficient across a stage,  c , and c2 corrected mass flow rate, m

 ci  m

 i i m

i 

i m i 

Pti Tti

P Tti  T ref Pref

Ai  MFP( M i )

• plots like these help engineers determine appropriate matching of compressor mass flow rate and pressure rise across a stage and what efficiencies you can expect from various combinations - these were created from lots of test geometries and conditions Mark McQuilling

ESCI 3200 Fluid Dynamics

v 1.2

Saint Louis University

Slide 7

Nature of dimensional analysis (3) • this plot shows pressure rise versus efficiency across a combustor section as functions of (dimensional) temperature rise; these plots have a lot of extremely useful information about how the engine changes its performance with changes the pilot can make! (Mach number, altitude, and throttle setting or Tt4; rotation speed is set by on-board computers)

• this one at left shows a turbine performance map and combinations of mass flow rate and pressure decrease (~ equivalent to work extracted) that will allow you to stay away from choked flow (maximum limit on mass flow)

Mark McQuilling

ESCI 3200 Fluid Dynamics

v 1.2

Saint Louis University

Slide 8

4

10/30/2018

Buckingham Pi Theorem (1) We saw before how the drag force F on a sphere in cross flow depends on the sphere diameter, D , fluid density  and viscosity  , and fluid speed V , which we can write as: F  f ( D,  ,  , V )

or

g ( F , D, , , V )  0

• Buckingham Pi Theorem states we can transform a relationship between n independent variables into one between n – m independent dimensionless  parameters as below:

g( q1, q2 ,..., qn )  0

G( 1,  2 ,...,  n m )  0

or

 1  G 1 ( 2 ,...,  n m )

• experiments help determine the functional form of the relationship (i.e. polynomial coefficients, etc.), and combinations of  groups as shown below are not independent: 5 

2 1  2 3

6 

13 / 4  23

- even though they may not be independent, they may still be useful Mark McQuilling

ESCI 3200 Fluid Dynamics

v 1.2

Saint Louis University

Slide 9

Buckingham Pi Theorem (2) There’s a six step procedure to produce the dimensionless groups: 1. List all dimensional parameters involved - the number of them is n - those included having no effect on the physical phenomena will either produce extra non-useful groups or they will not enter useful relations 2. Select primary dimensions: MLt, or FLt (T may also be needed for temperature, etc.) - the number of them is r - use either force F or mass M, but not both 3. List dimensions of all parameters in terms of primary dimensions 4. Select a set of r dimensional parameters that include all primary dimensions - these are called repeating parameters, and are used to non-dimensionalize groups 2 4 - repeating parameters must not be a power of each other (i.e. L and L ) - try to use viscosity, density, freestream velocity, and characteristic length if possible 5. Set up dimensional equations, combining parameters from Step 4 with each other parameter in turn, creating the non-dimensional  groups – there should be n – m equations 6. Verify that each new group is dimensionless (it’s easy to make a mistake) Mark McQuilling

ESCI 3200 Fluid Dynamics

v 1.2

Saint Louis University

Slide 10

5

10/30/2018

Buckingham Pi Theorem (3) Example problem 22 (Example 7.1): Obtain a set of dimensionless groups that can be used to correlate experimental wind tunnel drag measurements on a sphere. (This example was introduced as Example 2 early in the semester to help prepare you for Fluids Lab, but we’ll go over it again here for clarity.)

Mark McQuilling

ESCI 3200 Fluid Dynamics

v 1.2

Saint Louis University

Slide 11

Buckingham Pi Theorem (4) Example problem 22 (continued):

Mark McQuilling

ESCI 3200 Fluid Dynamics

v 1.2

Saint Louis University

Slide 12

6

10/30/2018

Buckingham Pi Theorem (5) Here’s an example of how this same process was very important to my own academic experience – in 2005 I programmed a laminar to turbulent transition model into a 2D NASA flow solver in order to design a low pressure turbine airfoil that takes advantage of turbulence properties (mixing!) to perform better at low Reynolds numbers (i.e. high altitudes). • figure shows two transition models, both built using the Buckingham PI process: one for separated flow transition (left, 47 test cases), and the other for attached flow transition (right, 57 test cases)

McQuilling, “Design and Validation of a High-Lift Low-Pressure Turbine Blade,” PhD D issertation, Wright State University, 2007

• predicts when transition occurs for both situations based on local flow parameters

• the curve-fitted model is used in the CFD process by using a laminar flow solver in the domain before the condition is met, and using a turbulent flow solver after the condition is met - allows more accurate simulation = improved prediction = improved design Mark McQuilling

ESCI 3200 Fluid Dynamics

v 1.2

Saint Louis University

Slide 13

Similitude and model scaling (1) Scaled models used in wind tunnels yield data (flow parameters, forces, moments, and dynamic loads) that can be scaled to represent the data on the full-scale prototype. To achieve this, several types of similarity must occur between the model and the prototype. • geometric similarity requires the model and prototype be the same shape, and all linear dimensions of the model be related to the prototype by a constant scale factor • kinematic similarity requires velocities at corresponding points to be in the same direction but scaled by a (different) constant scale factor - kinematic similarity requires geometric similarity - kinematic similarity is normally compromised in wind tunnels due to their finite test section size that cannot approximate open space, but there are known correction factors to account for these “blockage effects” • dynamic similarity requires identical types of forces that are parallel and related in magnitude by a (different) constant scale factor at all corresponding points - kinematic similarity is a necessary (but not sufficient) condition for dynamic similarity - correction factors can correct for slight non-match mentioned above - all types of forces must be present for both the model and the prototype! Mark McQuilling

ESCI 3200 Fluid Dynamics

v 1.2

Saint Louis University

Slide 14

7

10/30/2018

Similitude and model scaling (2) When the similarity conditions are met, we can relate model data to prototype data only when the appropriate dimensionless  groups are matched. For example, the Buckingham Pi Theorem predicted the functional relation: F

 V 2 D2

  VD   f1    

• this means the group on the LHS will match between the model and the prototype as long as the group on the RHS matches as well: If

 VD   VD          model   prototype

 F   F    2 2   V 2D 2   model   V D prototype

then:

- the forces aren’t equivalent but the nondimensional groups are - different fluids (different  , ) can also be used as long as the group values match (see Example 7.4 on p257)

Mark McQuilling

ESCI 3200 Fluid Dynamics

v 1.2

Saint Louis University

Slide 15

Similitude and model scaling (3) Sometimes the physics involved forces an incomplete similarity, although reasonable understanding and estimates can still be gained. The book uses an example of modeling a surface ship, which would require matching Reynolds and Froude numbers, which requires: Frm 

Vp Vm  Frp  (gLm )1 / 2 (gL p )1 / 2

Rem 

Vm Lm

m

 Re p 

Vp Lp

p

3/ 2

 m  Lm    p  Lp 

• so a typical model scale factor of 1:100 requires a viscosity ratio of 1:1000, which cannot be achieved with known liquids; but useful information can still be generated • useful information can still be generated: for the surface ship test, total resistance data could be measured as seen in the figure at right, but other techniques would have to predict components of viscous and wave resistances (since both  groups were not matched) that comprise what is measured Mark McQuilling

ESCI 3200 Fluid Dynamics

v 1.2

Saint Louis University

Slide 16

8

10/30/2018

Similitude and model scaling (4) We’ve also already seen with our earlier performance maps of turbomachinery components that we can plot together multiple relationships on the same graph when there are common dimensionless (or dimensional) groups involved. These types of relationships exist for pump scaling laws, as shown below:

• Buckingham Pi Theorem results in the following for the flow coefficient, head coefficient, and power of a pump at constant rotational speed: Q1

1D13



Q2

h1

2 D23

 21 D12



h2

P1

 22D 22

 113 D15



P2

 223D 52

• different  groups exist, but these groups (and one for specific speed to get rotation speed effects) have proven sufficient for understanding performance of a wide range of pumps Mark McQuilling

ESCI 3200 Fluid Dynamics

v 1.2

Saint Louis University

Slide 17

Similitude and model scaling (5) Example problem 23 (Example 7.5): For the following drag force test data taken on a 1:16 scale model of a bus, plot the dimensionless drag coefficient versus Reynolds number where the model width and frontal area is 8ft and 84ft2, respectively. Find the minimum test speed above which CD remains constant, and estimate the aerodynamic drag force and power requirement for the prototype vehicle at 100km/hr.

Mark McQuilling

ESCI 3200 Fluid Dynamics

v 1.2

Saint Louis University

Slide 18

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10/30/2018

Similitude and model scaling (6) Example problem 23 (continued):

Mark McQuilling

ESCI 3200 Fluid Dynamics

v 1.2

Saint Louis University

Slide 19

Homework 6 2 • In flow at moderate to high Reynolds numbers, pressure scales with U  ; but in very viscous flows (low Re), pressure instead scales with U  / L. Re-do the non-dimensionalization of the x-momentum equation with this new scaling for pressure, define any parameters that arise, and show what happens if the Reynolds number is very small.

• 7.8 • 7.18 • 7.21 • 7.30 • 7.42 – part (b) use ideal gas law to compute pressure/density ratio, then Reynolds number to obtain velocity • 7.51 • 7.60 – see picture at right; build PI groups to include

,

and don’t include time in the Buckingham-PI process, calculate that at the end like this:

Mark McQuilling

ESCI 3200 Fluid Dynamics

v 1.2

Saint Louis University

Slide 20

10...