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9.1 Vis_04 Data Visualization Lecture 9 Vector Field Visualization - Visualizing Flow Part 1: Experimental Techniques Particle based Techniques

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9.2 Vis_04 Applications of Vector Field Visualization n The major application area is Computational Fluid Dynamics where we wish to visualize the velocity field within a volume, or on a surface flow n... but has applications to any other discipline where flow is involved – for example, the flow of population in social sciences steady unsteady n Important distinction between steady and unsteady (time-dependent) flow

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9.3 Vis_04 Virtual Windtunnel The Virtual Windtunnel is a VR facility developed by NASA for aircraft testing

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9.4 Vis_04 Visualizing Flow over Aircraft Wing

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9.5 Vis_04 Experimental and Computational Fluid Dynamics n Experimental fluid dynamics: – aim to get impression of flow around a scale model of object (eg smoke in wind tunnel) – disadvantages in cost, time and integrity n Computational fluid dynamics – simulation of the flow (Navier-Stokes equations) – visualization of the resulting velocity field so as to mimic the experimental techniques

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9.6 Vis_04 Experimental Flow Visualization - Adding Foreign Material n Time lines – row of small particles (hydrogen bubbles) released at right angles to flow - motion of line shows the fluid flow n Streak line – dye injected from fixed position for period of time - tracer of dye shows the fluid flow n Path line – small particles (magnesium powder in liquid; oil drops in gas) - velocity measured by photographing motion with known exposure time

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9.7 Vis_04 Experimental Flow Visualization - Other Techniques n Visualization of flow field on surface of object achieved by fixing tufts at several points on surface - orientation of threads indicates direction of flow Notice distinction: – tufts show flow past a fixed point (Eulerian) – bubbles etc show the flow from point of view of a floating object (Lagrangian)

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9.8 Vis_04 Experimental Visualization Example - Poster Paint in Water see www.flometrics.com

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9.9 Vis_04 Experimental Visualization - Particles Illuminated by Laser Sheet Light

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9.10 Vis_04 Computer Flow Visualization n Now look at methods for computer aided flow visualization n Assume initially velocity field given on 3D Cartesian grid velocity v x, v y, v z given at each grid point hexahedral cells

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9.11 Vis_04 Using Scalar Techniques n Sometimes it is useful to derive scalar quantities from the velocity field n For example, velocity magnitude – speed = sqrt (v x 2 + v y 2 + v z 2 ) n How would these be visualized?

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9.12 Vis_04 Arrows n Very simple technique n Arrow drawn at each grid point showing direction and size of velocity This works effectively enough in 2D

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9.13 Vis_04 Arrows n But in 3D it suffers from perception problems: Is it this? or this? Of course the picture quickly gets cluttered too

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9.14 Vis_04 Arrows n Arrows can be used successfully in 3D as follows: – by slicing the volume, and attaching arrows (with shadow effects) to the slice plane - this gives a hedgehog effect – by giving more spatial cues - drawing arrows as true 3D objects but clutter again a problem! {BTW - Eulerian or Lagrangian?}

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9.15 Vis_04 CFD simulation of laser example Flometrics - see www.flometrics.com

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9.16 Vis_04 Comparison of Experimental and Computational Visualization

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9.17 Vis_04 Tufts

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9.18 Vis_04 Particle Traces n This is analogous to experimental path lines - we imagine following the path of a weightless particle - cf a bubble n Suppose initial position - seed point - is (x 0, y 0, z 0 ) n The aim is to find how the path ( x(t), y(t), z(t) ) develops over time n Also called particle advection

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9.19 Vis_04 Particle Traces n Motion of a particle is given by: dx/dt = v x ; dy/dt = v y ; dz/dt = v z - three ordinary differential equations with initial conditions at time zero: x(0) = x 0 ; y(0) = y 0 ; z(0) = z 0 In 2D, we have: (x 0,y 0 ) (v x,v y ) is rate of change of position

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9.20 Vis_04 Particle Tracing - Numerical Techniques for Integrating the ODEs n Simplest technique is Eulers method: dx/dt = ( x(t+ t) - x(t)) / t = v x (p(t)) hence x(t+ t) = x(t) + t.v x (p(t)) n Similarly, for y(t) and z(t) In 2D, we have: (x 0,y 0 ) (x 1,y 1 ) ( t.v x, t.v y ) p=(x,y,z)

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9.21 Vis_04 Particle Tracing - Interpolation n As the solution proceeds, we need velocity values at interior points n (v x, v y, v z ) is calculated at current point (x,y,z) - by trilinear interpolation for example. In 2D, we have: (x 0,y 0 ) (x 1,y 1 ) (v x,v y ) found by interpolation (bilinear)

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9.22 Vis_04 Particle Tracing - Point Location n When we leave one cell, we need to determine which cell the new point belongs to (x 0,y 0 ) (x 1,y 1 ) -this is quite straightforward for Cartesian grids

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9.23 Vis_04 Particle Tracing - Algorithm find cell containing initial position {point location} whileparticle in grid determine velocity at current position {interpolation} calculate new position {integration} find cell containing new position {point location} endwhile

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9.24 Vis_04 Improving the Integration Eulers method is inaccurate (unless the step size t is very small) n Better is Runge-Kutta: – x* = x(t) + t.v x (p(t)) (and for y*, z*) – x(t+ t) = x(t) + t.{ v x (p(t)) + v x (p*)}/2 (and for y,z) n This is Runge-Kutta 2nd order - there is also a more accurate 4th order method n There is another source of error in particle tracing – what?

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9.25 Vis_04 Rendering the Particles - and Rakes n Particles may be rendered as – points - but are there better representations? n It is common to use a rake of seed points, rather than just one - rake can be line, circle, even an area...

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9.26 Vis_04 Particle Advection Example - Flow Around a Moving Car Created using IRIS Explorer

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9.27 Vis_04 Streak Lines and Time Lines n Streak lines – release continuous flow of particles for a short period n Time lines – release a line of particles at same instant and draw a curve through the positions at successive time intervals

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9.28 Vis_04 Stream Lines n Mathematically, stream lines are lines everywhere tangential to the flow n For a steady flow - what is the relation between stream lines and particle traces? stream lines

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9.29 Vis_04 Rendering Streamlines n In 3D, curves are hard to understand without depth cues n Ideas used include: – stream ribbons - each streamline drawn as a thin flat ribbon, showing twist; or two adjacent streamlines connected into ribbon, showing twist and divergence – tubes

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9.30 Vis_04 Streamlines Example Streamlines drawn as tubes - by K Ma of ICASE (see www.icase.edu)

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9.31 Vis_04 Steady Flow Visualization n Streamlines and stream ribbons best for flow direction n Particle traces best for flow speed n Note that derived quantities are also visualized: – flow speed as 3D scalar field – vorticity field as 3D vector field – vorticity magnitude as 3D scalar field (Vortex = rotational flow about axis vorticity = vector product of velocity and its gradient)

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9.32 Vis_04 Unsteady Flow Visualization n Recent research interest has been in the more complex case of unsteady flows, where velocity depends on time n Particle traces, streak lines and time lines can all be used n Streak lines seem to give the best results n Nice applet at: http://widget.ecn.purdue.edu/~meapplet/java/flowvis/Index.html

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9.33 Vis_04 Different Types of Grid n Rectilinear n Curvilinear n Unstructured

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9.34 Vis_04 Curvilinear Grids n Point location and interpolation are much harder than for Cartesian grids – a solution is to decompose each hexahedral cell into tetrahedra – point inside test then easier... – … and interpolation is linear n Point location – draw line to new point – calculate intersection with faces to determine adjacent tetrahedron – check whether point inside new tetrahedron

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