Modelling the two-dimensional motion of rigid ... - unizar.es

UNIVERSITA

DI PAVIA

Modelling the two-dimensional motion of rigid bodies in a flow: toward validation

E.Persi Universidad Zaragoza

The work is a collaborative effort between DICAr and CFM Group

Index

1. Modelling rigid bodies in a flow

2. Equation of motion for a sphere

3. Equations of transport in ORSA2D_WT

4. Validation of transport equation

(Sphere in uniform stationary flow, spherical projectile, water entry of a

sphere and of a spinning sphere)

4.1. Qualitative tests

5. Extension to cylinders

(Equations, existing experiments)

6. Conclusions and planned tests

11 Noviembre 2016 Seminarios del Grupo de

Mechanica de Fluidos 2




riverbank erosion

debris flow

How much wood?

Critical situations?

Effects on water level?

MANAGE & PREVENT

How to prevent?

2D numerical modelling of floating

rigid bodies

How to design retention structures?

Water + sediments…

…+ floating wood!

Effects on structures?

landslides Photo Comiti et al. 2016

Photo Comiti et al. 2016

Photo Comiti et al. 2016 Photo Aipo




Geobrugg, 2007

Torrente Rienza FHWA

Photo Comiti et al. 2016

Examples of retention structures


PhD research

• Modelling wood transport (WT) during floods with a dynamic approach

• Track the displacement and rotation induced by the flow on a cylinder on a plane (i.e. a log on the water surface)



Eulerian approach

SWE (2D, velocity in x and y direction)

Lagrangian approach

DEM (Discrete Element Modelling)

The model proposed includes translation and rotation equations

ORSA2D_WT




Study the transport of spheres GOAL: get the proper transport equations

Critical issues

• Logs are modelled as cylinders, not symmetrical

• Translation and rotation are simultaneous

• Separate validation only for trivial cases

• Log orientation affects transport

• Rotation formulation is doubtful

Let’s simplify…

• Early 1900: Basset-Boussinesq-Oseen examined the motion of a sphere settling in a fluid at rest (BBO equation)

• 1950-1990: Several authors working on the formulation (Tchen, Corrsin and Lumely, Luhillier, Maxey and Riley, Saffmann) and extended the BBO to the case of a small rigid sphere in unsteady and non-uniform flow, and including lift force




Force on the body Pressure gradient Forces due to added mass

𝐷𝑽𝑓

𝐷𝑡=𝜕𝑽𝑓

𝜕𝑡+ (𝑽𝑓∙ 𝜵)𝑽𝑓

Lagrangian derivative

𝑚𝑏𝑑𝑽𝑏𝑑𝑡= 𝑭𝑑𝑟𝑎𝑔 + 𝑭𝑙𝑖𝑓𝑡 +𝑚𝑓

𝐷𝑽𝑓

𝐷𝑡+ 1

2𝐶𝐴𝑚𝑓

𝐷𝑽𝑓

𝐷𝑡−𝑑𝑽𝑏𝑑𝑡+ 𝑭𝑡𝑖𝑚𝑒 ℎ𝑦𝑠𝑡𝑜𝑟𝑦 + 𝑭𝑏𝑢𝑜𝑦𝑎𝑛𝑐𝑦

Stokes flow expressions



The equation is valid for:

• Uniform or nonuniform flow, steady or unsteady flow (see previous slide), usually 3D

• Small sphere (i.e. smaller then the spatial variations of the undisturbed flow)

• Small particle Reynolds number (Rep, Stokes flow)

High Rep, with the proper force expression

• Rigid spheres

Deformable bubbles or droplets (e.g. Magnaudet and Eames 2000, drag coefficient depends on aspect ratio up to a certain deformability (Moore 1965); derived for bubbles rising in still liquid)

• Fully submerged bodies

𝑅𝑒𝑝 =𝑽𝑟𝑒𝑙 𝑑

ν





Our problem:

• 2D

• Several type of flow: uniform, nonuniform, steady, (unsteady)

• Finite spheres (i.e. in general NOT smaller then the spatial variations of the undisturbed flow) AND large cylinders

• Variable Rep: high when the body velocity is different from flow velocity, low when the body reaches the flow velocity

• Rigid solid bodies

• Bodies floating on the water surface




𝑚𝑏 +1

2𝐶𝐴𝑚𝑓

𝑑𝑽𝑏𝑑𝑡=1

2𝜌𝐶𝐷𝐴𝐷 𝑽𝑓 − 𝑽𝑏

2+1

2𝜌𝐶𝑆𝐴𝐷 𝑽𝑓 − 𝑽𝑏

2+𝑚𝑓 1 +

1

2𝐶𝐴𝐷𝑽𝑓

𝐷𝑡

• AD is the reference area, for spheres: 𝐴𝐷 = 𝜋𝑟2

• CA is the added mass coefficient, CD is the drag coefficient, CS is the side coefficient

• We disregard time history force (can we do so?)

• Buoyancy is only included when needed (i.e. test cases on a vertical plane)

Force on the body Drag force Side force Forces due to pressure gradient and added mass

What about coefficients for submerged spheres?

Added mass coefficient (CA): a function of the body shape; for a sphere, for the form of the equation, is equal to 1.

Drag coefficient (CD): depends on the particle Reynolds number

Side/lift coefficient (CS): zero for non-rotating sphere, non-zero for rotating sphere (Magnus effect)




𝑚𝑏 +1

2𝐶𝐴𝑚𝑓

𝑑𝑽𝑏𝑑𝑡=1

2𝜌𝐶𝐷𝐴𝐷 𝑽𝑓 − 𝑽𝑏

2+1


2+𝑚𝑓 1 +

1


𝐷𝑡

The equation has been implemented in ORSA2D_WT, and will be employed to simulate the transport of spheres and cylinders on the water surface (new application)

Before it, we need to calibrate it in standard conditions (submerged small sphere, comparison with literature results)



3. Equations of transport in ORSA2D_WT – CD

Rep CD

≤1 24/Re

>1, ≤400 24/(Re0.646)

>400, ≤3.0E5 0.5

>3.0E5, ≤2E6 0.000366Re0.4275

CD for a smooth sphere of dimeter d (Chow 1979)

For submerged bodies

Stokes flow interval



Side coefficient

𝐶𝑆 = 0 if ω = 0

if ω ≠ 0, with 𝐶𝑆∗ = 𝑓(𝑅𝑒, 𝜔) from

Truscott and Techet (2009)

Rep

x105

3. Equations of transport in ORSA2D_WT – CS

CS* 𝑆0 =𝝎0𝑟

𝑽0

𝐶𝑆 = 𝐶𝑆∗ 𝑠𝑔𝑛((𝑽𝑓−𝑽𝒃) × 𝝎)




1) Spherical projectile

From Chow (1979): d=0.05 m, ρp=8000 kg m-3, initial velocity Vb=50 m s-1 and angle 30°

Motion in still air, in the plane (x,z)

No rotation, no flow velocity

Gravitation and buoyancy are considered

x

z

Sketch of the problem




1) Spherical projectile

𝑚𝑏 +1

2𝐶𝐴𝑚𝑓

𝑑𝑽𝑏(𝑡)

𝑑𝑡

=1

2𝜌𝐶𝐷𝐴𝐷 𝑽𝑓 − 𝑽𝑏(𝑡)

2+1


2+𝑚𝑓 1 +

1


𝐷𝑡+ 𝒈(𝑚𝑓 −𝑚𝑏)

Comparison of the computed trajectory with the numerical simulation by Chow (1979)




2) Sphere moving in a uniform stationary flow

d=0.03 m, ρp=720 kg m-3, initial velocity Vb=0 m s-1

Flow horizontal velocity Vfx=0.5 m s-1

No rotation, no velocity gradients or time variation

Motion in the plane (x,y)

x

y





2) Sphere moving in a uniform stationary flow - results

𝑽𝑏 = 𝑽𝑓 1 −1

𝑽𝑓

12𝜌𝐶𝐷𝐴

𝑚𝑏 +12𝑚𝑓𝐶𝐴

𝑡 − 𝑡0 + 1

Comparison of the computed velocity with the analytical solution:

𝑚𝑏 +1

2𝐶𝐴𝑚𝑓

𝑑𝑽𝑏(𝑡)

𝑑𝑡=1


2+1


2+𝑚𝑓 1 +

1


𝐷𝑡




3) Water entry of a sphere

From Guthrie (1975): d=0.0107 m, ρp=716 kg m-3, initial velocity Vbz=-7.74 m s-1

Motion in still water, in the plane (x,z)

No rotation, no flow velocity


x

z





3) Water entry of a sphere

𝑚𝑏 +1

2𝐶𝐴𝑚𝑓

𝑑𝑽𝑏(𝑡)

𝑑𝑡

=1


2+1


2+𝑚𝑓 1 +

1



Comparison with experimental results by Guthrie (1975)

Sinking of the sphere VS time




4) Water entry of a spinning sphere

From Truscott and Techet (2009): d=0.0572 m, ρp=1735 kg m-3, initial velocity Vbz=-5.74 m s-1, angular velocity ω=267.67 rad s-1

Motion in still water, in the plane (x,z)

Yes rotation, no flow velocity


x

z





4) Water entry of a spinning sphere

𝑚𝑏 +1

2𝐶𝐴𝑚𝑓

𝑑𝑽𝑏(𝑡)

𝑑𝑡

=1


2+1


2+𝑚𝑓 1 +

1



Comparison of the computed trajectory with experimental results by Truscott and Techet (2009)

In this case, ω is the angular velocity of the sphere

𝐶𝑆 = 𝐶𝑆∗ 𝑠𝑔𝑛((𝑽𝑓−𝑽𝒃) × 𝝎)




5) Water in solid body rotation - ρp > ρf

Qualitative tests in a rotational vortex, with a sphere heavier then water (ρp/ ρf =1.45)

x

y

Stokes flow, fully submerged

Side force should be accounted for…

𝑚𝑏 +1

2𝐶𝐴𝑚𝑓

𝑑𝑽𝑏(𝑡)

𝑑𝑡=1


2+1


2+𝑚𝑓 1 +

1


𝐷𝑡



a) Trajectory of a water sphere, with the same initial velocity of the fluid;

b) Trajectory of an initially motionless water sphere;

c) Trajectory of a motionless sphere with a density lower than water density.

6) Water in solid body rotation - ρp ≤ ρf


Qualitative tests with different sphere density and initial velocity




Translation equation

𝑚𝑏 +1

2𝐶𝐴𝑚𝑓

𝑑𝑽𝑏𝑑𝑡= 1

2𝜌𝐶𝐷𝐴𝐷𝑖 𝑽𝑓𝑖 − 𝑽𝑏𝑖

24

𝑖=1

+ 1

2𝜌𝐶𝑆𝐴𝐷𝑖 𝑽𝑓𝑖 − 𝑽𝑏𝑖

24

𝑖=1

+𝑚𝑓 1 +1


𝐷𝑡

Total drag force Total side force

Cylinders are not symmetrical and are large with respect to the flow variations




Translation equation

𝑚𝑏 +1

2𝐶𝐴𝑚𝑓

𝑑𝑽𝑏𝑑𝑡= 1

2𝜌𝐶𝐷𝐴𝐷𝑖 𝑽𝑓𝑖 − 𝑽𝑏𝑖

24

𝑖=1

+ 1

2𝜌𝐶𝑆𝐴𝐷𝑖 𝑽𝑓𝑖 − 𝑽𝑏𝑖

24

𝑖=1

+𝑚𝑓 1 +1


𝐷𝑡

Total drag force Total side force Forces due to pressure gradient and added mass

CD measured at UniPV CS measured at UniPV

CA = 2 for fully submerged cylinders




Rotation equation

Offset angular momentum

𝐼𝑑𝝎𝑏𝑑𝑡= 𝑭𝐷 + 𝑭𝑆 × 𝒓𝐶𝑃 𝑖

4

𝑖=1

1)

The cylinder is divided in 4 parts, so forces are computed in each section

2) 𝐼𝑑𝝎𝑏𝑑𝑡= 𝒙𝐶𝑃 × 𝑭+𝑴𝑅

Resistance momentum

𝑴𝑅 = 𝜌𝑑 𝜔𝑓 − 𝜔𝑏2𝐿41

64+ 3.36

𝜌𝑑 𝜔𝑓 − 𝜔𝑏 𝐿

𝜇

23 −1

Mandø and Rosendahl 2013

Centre of pressure location (for the offset angular momentum) Cylindrical bodies settling in calm water



5. Extension to cylinders – validate translation

Case study: horizontal channel, Manning coeff. 0.01 m-1/3s, 6 lateral obstacles, constant discharge of 18 l/s, weir height 5.8 cm. (Ruiz-Villanueva et al., 2014, IBER)

0.13 m

0.18 m

7.00 m

0.60 m x

y



5. Extension to cylinders – validate translation (2)

log type 5: L=0.20m, d=0.018m (?)

Corr. Coeff(x)= 0.99905; Corr. Coeff(y)= 0.75567



5. Extension to cylinders – validate rotation

Rotation equation

Offset angular momentum

𝐼𝑑𝜔𝑏𝑑𝑡= 𝑭𝐷 + 𝑭𝑆 × 𝒓𝐶𝑃 𝑖

4

𝑖=1

1)

The cylinder is divided in 4 parts, so forces are computed in each section

2) 𝐼𝑑𝝎𝑏𝑑𝑡= 𝒙𝐶𝑃 × 𝑭+𝑴𝑅

Resistance momentum Centre of pressure location (for the offset

angular momentum)

Force balance

1) Simplified formula

2) From literature, but obtained for different condition



6. Conclusions (1)

• A model to predict the 2D transport of floating rigid bodies is presented: both translation and rotation must be taken into account

• To simplify the validation, we focus firstly on the translation of spheres (symmetrical, punctual)

• Motion equation from literature is implemented in the numerical model, disregarding some terms (history, buoyancy only if needed) and adopting the appropriate drag and side coefficients

• Lift/side force is included, and appears only when the sphere is rotating

• The comparison of the numerical results with analytical solution and experiments from literature allows to partially validate the translation formulation

• Some problems remains open: pressure gradient and added mass forces, exact formulation of side force



6. Conclusions (2)

• The translation equation is extended to the case of cylinders

• We take into account the distribution of velocity on the cylinder length

• Drag and side coefficients varies with cylinder orientation

• The formulation of rotation taken from literature is currently implemented, even if it was obtained for a different condition (sinking elongated bodies)

• A validation of the translation formulation for cylinder was realized against literature data

A complete validation is still missing, both for spheres and cylinders motion.

For this reason, laboratory tests will be realized at Unizar



6. Planned tests

Irrotational vortex:

Side curve obstacle:

Floating bodies in a channel:

x

y

x

y

UNIVERSITA

DI PAVIA

Universidad Zaragoza

Thank you for your attention! ¿Questions?

[email protected]

The work is a collaborative effort between DICAr and CFM Group

Modelling the two-dimensional motion of rigid ... - unizar.es

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