Download Ahadowgraph - Optical Measurement Techniques in Thermal Sciences - Lecture Slides and more Slides Mechanical Engineering in PDF only on Docsity!
Module 5: Schlieren and Shadowgraph
Lecture 26: Introduction to schlieren and shadowgraph
The Lecture Contains:
Introduction
Laser Schlieren
Window Correction
Shadowgraph
Shadowgraph Governing Equation and Approximation Numerical Solution of the Poisson Equation Ray tracing through the KDP solution: Importance of the higher-order effects Correction Factor for Refraction at the glass-air interface Methodology for determining the supersaturation at each stage of the Experiment
Module 5: Schlieren and Shadowgraph
Lecture 26: Introduction to schlieren and shadowgraph
Introduction
Closely related to the method of interferometry are and that employ variation in refractive index with density (and hence, temperature and concentration) to map a thermal or a species concentration field. With some changes, the flow field can itself be mapped. While image formation in interferometry is based on changes in the the refractive index with respect to a reference domain, schlieren uses the transverse derivative for image formation. In shadowgraph, effectively the second derivative (and in effect the Laplacian ) is utilized. These two methods use only a single beam of light. They find applications in combustion problems and high-speed flows involving shocks where the gradients in the refractive index are large. The schlieren method relies on beam refraction towards zones of higher refractive index. The shadowgraph method uses the change in light intensity due to beam expansion to describe the thermal/concentration field.
Before describing the two methods in further detail, a comparison of interferometry (I), schlieren (Sch) and shadowgraph (Sgh) is first presented. The basis of this comparison will become clear when further details of the measurement procedures are described.
1. Interferometry relies on the changes in the refractive index in the physical region and hence
the changes in the optical path length relative to a known (reference) region. Schlieren measures the small angle of deflection of the light beam as it emerges from the test section. Shadowgraph measures deflection as well as displacement of the light beam at the exit plane of the apparatus.
2. Displacement effects of the light beam are neglected in schlieren while displacement as well as
deflection effects are neglected in interferometry. In effect, the light rays are taken to travel straight during interferometry.
3. Since large gradients will displace and deflect the light beams, interferometry is suitable for
small gradients and shadowgraph for very large gradients. Schlieren fits well in the intermediate range.
- In a broad sense, interferometry yields the refractive index field , schlieren - the gradient field and shadowgraph.
- Since deflection and displacement calculations are more complicated than that of the optical path length, shadowgraph analysis is the most involved, schlieren is less so, and interferometry is the simplest of the three.
- All the three methods yield a cumulative information of the refractive index field (or its gradients), integrated in the viewing direction, i.e. along the path of the light beam.
- As will be seen in the text of this module, schlieren and shadowgraph methods require simpler optics than interferometry. Shadowgraph is the simplest of all. The price to be paid is in terms of the level (and complexity) of analysis.
Module 5: Schlieren and Shadowgraph
Lecture 26: Introduction to schlieren and shadowgraph
In a schlieren measurement, beam displacement normal to the knife edge will translate into an intensity variation on the screen. Displacements that are blocked by the knife edge sheet are not recorded. Similarly, displacements parallel to the knife edge will also not change the intensity distribution. Information about these gradients in the respective directions can be retrieved by suitably orienting the knife edge. Other strategies such as using a gray scale filter are available. A color filter leading to a color schlieren measurement is desrcibed later in this module.
Consider the displacement of ray 1 as in Figure 5.2. At point P the illumination is proportional to a , say, equal to. With the test cell in place this becomes^. Hence at^ the contrast with respect to the undisturbed region is proportional to. The contrast increases greatly when the initial illumination is small, but it can lead to difficulties in recording the schlieren pattern. It can be shown that
and the contrast can be adjusted using the focal length of lens.
Figure 5.2: Initial and final positions of the light spot with respect the knife
edge.
Module 5: Schlieren and Shadowgraph
Lecture 26: Introduction to schlieren and shadowgraph
Laser Schlieren
Image formation in a schlieren system is due to the deflection of light beam in a variable refractive index field towards regions that have a higher refractive index. In order to recover quantitative information from a schlieren image, one has to determine the cumulative angle of refraction of the light beam emerging from the test cell as a function of position in the plane. This plane is defined to be normal to the light beam, whose direction of propagation is along the coordinate.The path of the light beam in a medium whose index of refraction varies in the vertical direction can be analyzed using the principles of geometric or rays optics as follows:
Consider two wave fronts at times and as shown in Figure 5.3. At time the ray is at a position. After a interval , the light has moved a distance of times the velocity of light, which in general, is a function of , and the wave front or light beam has turned an angle. The local value of the speed of light is where is the velocity of light in vaccum and is the refractive index of the medium. Hence the distance that the light beam travels during time interval is
There is a gradient in the refractive index along the direction. The gradient in results in a bending the wave front due to refraction. The distance is given by
Let represent the blending angle at a fixed location. For a small increment in the angle, can be expressed as
In the limiting case
(1)
Hence the cumulative angle of the light beam at the exit of the test region will be given by
(2)
where the integration is performed over the entire length of the test region.
The dotted lines shows the path of the light beam in the presence of disturbances in the test region. The second lens , kept at the focus of the knife-edge collects the light beam and passes onto the screen located at the conjugate focus of the test section.
Module 5: Schlieren and Shadowgraph
Lecture 26: Introduction to schlieren and shadowgraph
Figure 5.4: View of deflected and undistributed beams at the knife-edge of a
schlieren system.
If no disturbance is present, the light beam at the focus of would be ideally as shown in Figure 5.4, with dimensions. These are related to the initial dimensions by the formulas
where and are the focal lengths of and , respectively.
In a schlieren system, the knife edge kept at the focal length of the second concave mirror is first adjusted, when no disturbance in the test region is present, to cut off all but an amount correnponding to the dimension of the light beam Let be the original size of the laser beam.If the knife edge is properly positioned, the illumination at the screen uniformaly changes depending upon its direction of the movement. Let be the be the illumination at the screen when no knife- edge is present. The illumination with the knife-edge inserted in the focal plane of the of the second concave mirror but without any disturbance in the test region will be given by
(4)
Let be the deflection of the light beam in the vertical direction above the knife edge corresponding to the angular deflection ( ) of the beam after the test region experiences a change in the refractive index. Then from Figure 5.4, can be expressed as
(5)
where the sign is determined by the direction of the displacement of the laser beam in the vertical direction; it is positive when the shift is in the upward direction and negative if the laser beam gets
Module 5: Schlieren and Shadowgraph
Lecture 26: Introduction to schlieren and shadowgraph
Contd...
Let be the final; illumination on the screen after the light beam has deflected upwards by an amount due to the inhomogeneous distribution of refractive index gradients in the test cell. Hence
(6)
The change in the light intensity on the screen is given by
The relative intensity or contrast can be expressed as
(7)
Using Equation 5
(8)
Equation 8 shows that the contrast in a schlieren system is directly proportional to the focal length of the second lens. Larger the focal length, greater will be the sensitivity of the system.
Combining Equations 3 and 8
(9)
This equation shows that the schlieren technique records the average gradient of refractive index over the path of the light beam. If the field is two dimensional with the refractive index gradient constant at a given position over the length in the direction, then
(10)
Module 5: Schlieren and Shadowgraph
Lecture 26: Introduction to schlieren and shadowgraph
Equation 10 holds for every position in the test section and gives the contrast at the equivalent position in the image on the screen. The quantity on the left hand side can be obtained by using the initial and final inensity distribution on the screen. is the length of the test section along the direction of the propogation of the laser beam, is the focal length of the second concave mirror and is the size of the focal spot at the knife-edge. Usually, the knife-edge is adjusted in such a position that it cuts off approxiamtely 50% of the original light intensity, i.e. where is the original dimension of the laser beam at the pin-hole of the spatial filter. Typically, for a He-Ne laser (employed as the light source in the present work) is of the order of 10-100 microns. With Equation 10 can be written as
(11)
Equation 11 represents the governing equation for the schlieren process in terms of the ray-averaged refractive index. It requires the approximation that changes in the light intensity occur due to beam deflection, rather than its physical displacement.
If the working fluid is a gas (e.g. air as employed in the validation experiments of the present study), the first derivative of the refractive index field with respect to y can be expressed as
(12)
Equation 12 relates the gradient in the refractive index field with the gradients of the density field with the gradients of the density field in the fluid medium inside the test cell. The governing equation for the schlieren process in gas (Equation 11) can be rewritten as
(13)
Assuming that pressure inside the test cell is practically constant, we get
(14)
Equation 13 and 14 respectively relate the contrast measured using a laser schlieren technique with the density and temperature gradients in the test section. With the value of the dependent variables defined in the bulk of the fluid medium or with proper boundary conditions, the above equations can be solved to determine the quantity of interest.
Module 5: Schlieren and Shadowgraph
Lecture 26: Introduction to schlieren and shadowgraph
Window Correction
For visualization of the concentration field by the schlieren technique, circular optical windows have been fixed on the walls of the growth chamber at opposite ends. THe optical window employed in the present discussion of crystal growth is of finite thickness (5 mm) and the index of refraction of its material (BK-7) is considerably different from that of a KDP solution and air. The light beam emerging out of the KDP solution with an angular deflection due to the variable concentration gradients in the growth chamber again undergoes refraction before finally emerging into the surrounding environment. The contribution of refraction of light at the confining optical windows can be accounted for by applying a correction factor in Equation 11 as discussed below.
The laser beam strikes the second optical window fixed on the growth chamber at an angle after undergoing refraction due to variable concentration gradients in the vicinity of the growing KDP crystal. The optical windows are made of BK-7 material with index of refraction equal to 1.509. The refractive index of the KDP solution at an average temperature of is equal to 1.355 and for air. Let be the angular deflection of the beam purely due to the presence of concentration gradients in the vicinity of the growing crystal as shown schematically in Figure 5.6.
Figure 5.6: Schematic drawing showing the path of the light beam and the
corresponding angles of deflection as it passes through the growth chamber.
(Dimension in the figure are exaggerated for clarity).
The beam strikes the second optical window at this angle. Let be the angle at which the laser beam emerges out of the second optical window. The angle at which the laser beam emerges out of the second optical window can be determined in terms of using Snell's law as
(16)
Module 5: Schlieren and Shadowgraph
Lecture 26: Introduction to schlieren and shadowgraph
Contd...
Since is quite small, , and
(17)
Let be the final angle of refraction with which the laser beam emerges into the surrounding air. For the optical window-air combination,
(18)
Substituting the value of from Equation 17 into Equation 18, the angle with which the laser beam emerges into the surrounding medium can be expressed as
(19)
or
(20)
Since
(21)
Hence a correction factor equal to the refractive index of the KDP solution at the ambient temperature is taken into consideration for calculating the angle at which the laser beam emerges into the surrounding medium.
Module 5: Schlieren and Shadowgraph
Lecture 26: Introduction to schlieren and shadowgraph
Contd...
Here is the change in illumination on the screen due to the beam displacement from its original path and is the original intensity distribution. Equation 22 implies that the shadowgraph is sensitive to changes in the second derivative of the refractive index along the line of sight of the of the light beam in the fluid medium. Integration of the Poisson equation (22) can be performed by a numerical technique, say the method of finite differences.
From Equation 22 it is evident that the shadowgraph is not a suitable method for quantitative measurement of the fluid density, since such an evaluation requires one to perform a double integration of the data. However, owing to its simplicity the shadowgraph is a convenient method for obtaining a quick survey of flow fields with varying fluid density. When the approximations involved in Equation 22 do not apply, shadowgraph can be used for flow visualization alone.
The present lecture has a discussion on how shadowgraph images can be analyzed to retrieve information on the concentration field.
Shadowgraphy
It employs an expanded and collimated beam of laser light. The light beam traverses the field of disturbance, an aqueous solution of KDP in the present application. If the disturbance is a field of varying refractive index, the individual light rays passing through the field are refracted and bent out of their original path. This causes a spatial modulation of the light intensity distribution. The resulting pattern is a shadow of the refractive-index field in the region of the disturbance.
Governing equation and Approximations
Consider a medium with refractive index that depends on all the three space coordinates, namely
. We are interested in tracing the path of light rays as they pass through this medium. Starting with the knowledge of the angle and the point of incidence of the ray at the entrance plane, we would like to know the location of the exit point on the exit window, and the curvature of the emergent ray.
Let the ray be incident at point and the exit point be^. According to Fermat's principle the optical path length traversed by the light beam between these two points has to be an extremum. If the geometric path length is , then the optical path length is given by the product of the geometric path length with the refractive index of the medium. Thus
(23)
Module 5: Schlieren and Shadowgraph
Lecture 26: Introduction to schlieren and shadowgraph
Parameterizing the light path by , the condition (Equation 23) can be represented by two functions , and the equation can be written as
(24)
where the primes denote differentiation with respect to. Application of the variational principle to the above equation yields two coupled Euler-Lagrange equations, that can be written in the form of the following differential equations for :
(25)
(26)
The four constants of integration required to solve these equations comes from the boundary conditions at the entry plane of the chamber. These are the co-ordinates of the entry point and the local derivatives. The solution of the above equation yields the two orthogonal components of the deflection of the ray at the exit plane, and also its gradient on exit.
In the experiments performed, the medium has been confined between parallel planes and the illumination is via a parallel beam perpendicular to the entry plane. The length of the growth chamber containing the fluid is D and the screen is at a distance behind the growth chamber. The coordinates at entry, exit and on the screen are given by respectively. Since the incident beam is normal to the entrance plane, there is no refraction at the optical window. Hence the derivatives of all the incoming light rays at the entry plane are zero;. The displacements of the light ray on the screen with respect to its entry position are
(27) (28)
where are given by the solutions of previous equations at.
The above formulation can be further simplified with the following assumptions.
Module 5: Schlieren and Shadowgraph
Lecture 26: Introduction to schlieren and shadowgraph
Assumption 2 : The assumption of the infinitesimal displacement inside the growth chamber can be extended and taken to be valid even for the region falling between the screen and the exit plane of the chamber. As a result, the coordinates of the ray on the screen can be written as
(38) (39)
The deviation of the rays from their original paths in occurs through the inhomogeneous medium. In the absence of the inhomogeneous field, such an area is a regular quadrilateral. It transforms to a deformed quadrilateral when imaged on to a screen in the presence of the inhomogeneous field. The summation in the above equation extends over all the rays passing through points at the entry of the test section that are mapped onto the small quadrilateral on the screen. Considering the fact that the area of the initial spread of the light beam gets deformed on passing through the refractive medium, the intensity at point is
(40)
where is the intensity on the screen in the presence of the inhomogeneous refractive index field, and is the original undisturbed intensity distribution. The denominator in the above equation is the Jacobian of the mapping function of onto as shown in Ffigure 5.8.
Figure 5.8: Jacobian of the mapping function onto
Geometrically it represents the ratio of the area enclosed by four adjacent rays after and before passing through the inhomogeneous medium. In the absence of the inhomogeneous field, such an area is a regular quadrilateral. It transforms to a deformed quadrilateral when imaged on to a screen in the presence of the inhomogeneous field. The summation in the above equation extends over all the rays passing through points at the entry of the test section that are mapped onto the small quadrilateral on the screen and contribute to the light intensity within.
Module 5: Schlieren and Shadowgraph
Lecture 26: Introduction to schlieren and shadowgraph
Assumption 3: Under the assumption of infinitesimal displacements, the deflections are small. Therefore the Jacobian can be assumed to be linearly dependent on them by neglecting the higher powers of , and also their product. Therefore, the jacobian can be expressed as
(41)
Substituting in Equation 39, we get
(42)
Simplifying further we get
(43)
Using Equations 36 and 37 for , and integrating over the dimensions of the growth chamber, we get
(44)
Equation 44 is the governing equation of the shadowgraph process under the set of linearizing approximation 1-3. In concise from the above equation can be rewritten as
(45)
