\begin{spacing}{0}
\chapter{Experimental Investigation of Mercury Jet Flow in a Magnetic Field}
\label{chap:exp_result_mhd}
\end{spacing}
\
\par

In this chapter, Hg jet behavior in a magnetic field is investigated. To do this, the collected images are read digitally and the characteristic jet parameters are evaluated based on measurement. 
Jet deformation such as the free jet surface deformation and surface stabilization 
are investigated by measuring pixels on the collected images based on 2-D shadow photography. 
Magnetic field effect on the dynamic behavior of freely moving jet in a magnetic field is discussed.
The driving pressure of Hg flow entering inlet pipe is measured to monitor the effect of
the magnetic field and assure if the input condition for driving the jet is affected.
In order to diagnose the flow rate, the flow velocity in magnetic fields is discussed and the
deflection of jet size in various magnetic field is investigated. Based on the observed flow
rate of jet, the shape of jet is suggested for the energy deposition calculation by proton beam
interaction with Hg jet target.
\\

\section{Image Analysis for Data Reduction}

\subsection{Image acquisition}

$\sim $ 360 complete integrated tests (i.e., with magnet, proton beam, Hg loop system, and optical diagnostic system) were conducted at CERN (European Organization for Nuclear Research) with various values of the proton beam structure (8 harmonic and 16 harmonic), the beam intensity up to
30 $\times$ 10$^{12}$ protons, the beam energy (14~Gev and 24~GeV), and the the magnetic field (0~T, 5~T, 7~T, 10~T, and 15~T) with 15~m/s Hg jet velocity. Figure~\ref{fig:jet15ms} and ~\ref{fig:jet20ms} are representative optical diagnostic results collected by the 3 cameras, with and without a magnetic induction field at Plasma Science
and Fusion Center in Massachusetts Institute of Technology. Note that only 15~m/s Hg jet shot was run at CERN,
and both 15~m/s and 20~m/s shot were run at PSFC in MIT. Also, note that the Olympus Encore PCI 8000S camera for Viewport 4 was integrated in the beam interacting target study done at CERN.

The current in the magnet system generates heat, which is cryogenically removed using liquid nitrogen. As the magnet cools down, all Viewports become foggy up due to condensation. It was found out that $\sim$ 0.5~$\ell$ of water (from nozzle performance test at Oak Ridge National Laboratory) was not removed from the system prior to loading Hg.
Flexible heater strips were installed both on the exterior of the primary containment vessel and on the snout in order to prevent the condensation of the humid air on the Viewports. Although residual Hg droplets in sizes less than 1~mm often adhere to the sapphire Viewports after every shot, jet motion with adequate image quality could still be collected.

\subsection{Image processing}

To measure the shape of the jet, 8 and 12 bit grey scaled TIF images are converted into digital forms. Background images are subtracted to reject the noise in the image digitization
process. The residual data is then transformed into a 2 bit scaled image.
Figure~\ref{fig:image_convert} shows the collected image and its transformed 2 bit scaled image.
Only the black and white colored pixels in the 2 bit depth images are used to differentiate
the shadow of the jet and the background.
Due to the image quality caused by the Hg droplet on window and the quality in fiber optic system,
the noise such as black dots exits. A threshold is adjusted according to Otsu's method to
highlight the interface between the Hg and background \cite{Otsu}.
Otsu's method selects the threshold by minimizing the within-class variance and
maximizing the between-class variance of the two groups of pixels separated by the thresholding operator.
Otsu's method, which relies on the assumption that all image pixels belong to one of two classes,
background or foreground, has been shown to be efficient in image segmentation for bi-level thresholding.

Figure~\ref{fig:data_example}(a) shows the sensitivity of 2 bit scaled image conversion to
the measurement of jet height with respect to the level of 8 bit threshold using Otsu's method.
As the threshold level increases, the mean
value of the jet height as well as the $\sigma$ value of the jet height in measurement is
approaching an asymptotic level. The optimally selected threshold value by the Otsu's method
in this example is 0.35.

The Hg jet was observed at upstream (Viewport 1), midstream (Viewport 2), and downstream (Viewport 3) locations from the nozzle exit. 220 images are collected at each run for
both the upstream and downstream locations, with an image size of 1280 $\times$ 1000 pixels.
The most probable transverse jet height within the longitudinal pixel range of 300 to 1000 is shown in the histogram of Fig.~\ref{fig:data_example}(b).
Note that within this range, the transverse jet height probability P could be obtained by
counting the number of longitudinal pixel events in the jet image.
Let $z$ denotes the transverse direction (in terms of pixels).
The number of background events (i.e., outside of the jet) is always larger than that within the jet
because the portion of bright background on each image is larger than that of the black jet shadow.
The distribution on the left in Fig.~\ref{fig:data_example}(b) (i.e., 0 $<$ $z$ $<$ 200) represents
the background pixels.
Then, the number of pixels corresponding to the jet height is counted within the longitudinal pixel
range of 300 to 1000. Each counted pixel numbers are directly averaged to give a jet height measurement and
then added up over $\sim$ 200 images for 1 shot of jet run, where the time elapse for the images corresponds
to $\sim$ 0.4~s at Viewport 1 and 3.
Multiple shots are then used to add up all of the counted vertical jet height. The average of the
individually counted vertical pixels is given to indicate the nominal jet height. In a mathematical form,
the direct averaging method is described as follow:

\begin{equation}
 \label{e:eq41a}
D_{jet} = \frac{1}{i+j+k} \sum_{1}^{i} \sum_{1}^{j} \sum_{1}^{k}  N_ {vertical} \ ,
\end{equation}

where $D_{jet}$ and $N_{vertical}$ denote the averaged vertical jet height and a individually counted
number of vertical pixels respectively. $i, j, k$ represent the number of shots, images in a shot, and
vertical lines in a image respectively.
On Viewport 2, 16 image files are collected at each run, with an image size of 316 $\times$ 316 pixels. The images are analyzed in the same manner as described above.
Viewports 1 and 3 give the same resolution for the images: 1280 $\times$ 1000.
Thus, no image re-scaling is needed when comparing the pixel size for these images.
However, Viewport 2 gives a resolution of 316 $\times$ 316. Based on the 1~cm scale fiducial mark on the exterior of all Viewports, all images taken on this Viewport are re-scaled to match the resolution of Viewport 1 prior to comparison.

\subsection{Study on the scaling length and the location of window center}

In order to relate the lengths on the collected images at each Viewport, the pixel length on the
images is investigated. Since the image size corresponds to the CCD size, any discrepancy in
horizontal and vertical pixel size is not considered.
Viewports 1 and 3 give the same resolution for the images: 1280 $\times$ 1000.
Thus, no image re-scaling is needed when comparing the pixel size for these images but we did
the scaling to see any difference on the image length of Viewport 1 and Viewport 3.
The fiducial length on the top front window and the bottom back window is measured and then
interpolated to get the length at the mid-span on the primary containment.
The interpolated pixel length at the mid-span corresponds to 1~cm at the mi-span of
primary containment. Thus, in Viewport 3, a pixel length at the mid-span where the jet is moving is
approximated $\sim$ 0.05~mm. Same scaling was done at images in Viewport 3.
The ratio of the pixel length in Viewport 3 to Viewport 1 is 1.06.

Viewport 2 gives a resolution of 245 $\times$ 252. Based on the 1~cm scaled fiducial mark on the exterior of all Viewports, all images taken on this Viewport are re-scaled to match the resolution of Viewport 1. 
A pixel length at the mid-span is approximated $\sim$ 0.21~mm.
Viewport 4 gives a different resolution of images depending on the frame rate setting but typically the resolutions of 320 $\times$ 280 was used. A pixel length at the mid-span is approximated $\sim$ 0.21~mm, which is same with Viewport 2.

The distance of the center position between the fiducial and the window is 0.75 inch apart.
In order to locate the center of the window at the mid-span, the position where 0.75~inch is apart
from the top fiducial and bottom fiducial is found on each image and then the averaged difference in
the located position is considered as the center of window.

Based on these scaling study, the measurement is performed in the following investigation.
The average and standard deviation of measurement are used to represent results of the 
following investigation. 
Based on the standard deviation and the number of events, the error bar, $\sigma / \sqrt{N}$,
is calculated to typically give error estimation for each measurement.

\section{Motion of Mercury Jet and Stability in a Magnetic Field}

\subsection{Jet deflection and surface flattening}

Jet motion in a magnetic field is expected to behave differently, depending on the angle between the axis
of magnet and the axis of jet, as a result of the differences in the magnitude of components of
magnetic field \cite{Samulyak2006}.
Figure~\ref{fig:magnet_field}(a) and (b) show the axial and radial components of the
magnetic field in a solenoid.  Figure~\ref{fig:magnet_field}(c) and (d) show the transverse and longitudinal components of the magnetic field along the jet axis at different crossing angles. As the crossing angle increases, the transverse component of the magnetic field increases, but with no significant change in the longitudinal component of the magnetic field. An increase of the transverse component of the magnetic field raises the induced axial current on the Hg jet. Therefore, the angle of the Hg jet is launched at 33~milliradian with respect to the axis of solenoid magnet. When the jet is injected without an applied magnetic field,
it is difficult to discern the jet surface because of blockage by Hg droplets on the window.
Therefore, some errors in the measurement exists (see images in Fig.~\ref{fig:jet15ms}(a) through Fig.~\ref{fig:jet15ms}(c) and ~\ref{fig:jet20ms}(a) through ~\ref{fig:jet20ms}(c)).
On the contrary, when a magnetic field is applied, the measurement errors are significantly
reduced,leading to significantly less intermittent jet boundaries.

The inertial forces appear to dominate the jet movement when the jet velocity is 15~m/s.
The turbulent jet motion is unstable but becomes stabilized as the magnetic field approaches 5~T.
It has been reported that the radial force induced by the transverse component of magnetic field
caused by the axially induced current due to the tilted jet angle can significantly increase
the jet height \cite{Gallardo2002}. The phenomena of increasing jet thickness
with high magnetic induction field is observed for the first time when the magnetic field exceeds 10~T.

Figure~\ref{fig:jet_height_reduced} shows the jet height measurement by direct averaging of vertical jet
height from measured pixels on each image.
The standard deviation is used to give the error bar. This two plot shows the extreme two conditions of
evaluation of the measured jet height, but one can effectively observe the fluctuating amount relative to
the nominal jet height according to the various magnetic fields.

At a jet velocity of 15~m/s, the relatively low inertial force reduces the extent of turbulent fluctuation.
For this case, the magnetic field does not significantly affect the dynamics of the jet
until the magnetic field strength of $\sim$ 5~T reaches. Consequently, the height of the jet
decreases only slightly until 5~T since the magnetic field reduces the fluctuating surfaces and the
jet is more likely to elongate axially to the jet axis.
The results shown in Fig.~\ref{fig:jet15ms} and ~\ref{fig:jet20ms} clearly suggest that
the magnetic field has constrained (stabilized) the Hg jet flow by smoothing out
the edges of the otherwise turbulent flow. At large number of the magnetic field ($>$10~T),
stability is maintained at all Viewports. At 15~T, a larger height (cross sectional distortion)
is observed on all Viewports.

The fact that the Hg jet size is relatively reduced from 0~T to 5~T but increases from 10~T to 15~T suggests that the Hg jet might encounter a different type of instability at high magnetic field, namely a quadrupole effect. The quadrupole effect would alter the jet's circular cross-section to become elliptical. From the data obtained with a 15~m/s jet, the jet height at 10~T is smaller than that at
15~T, which is manifested in the vertical elongation of the jet. However, the height at 10~T is
smaller than that at 5~T. The issues for such a behavior have to be addressed. There are two
possibilities. First, the jet is elongating axially up to 10~T. The equivalence of hydrodynamic
pressure with magnetic pressure is more dominantly affecting to the axial elongation of jet than the
transverse pressure.
However, the transverse magnetic pressure becomes significant once
the magnetic field exceeds 10~T. Thus, the jet at 15~T is experiencing the transverse deflection as
well as axial deflection, but the the role of transverse deflection plays significantly on the
behavior of jet. That can explain why the reduction of jet is appearing up to 10~T and then the
expansion of jet is appearing at 15~T.

Second, the optical diagnostics depends only on the side sectional view of jet movement.
The reduction of jet size on the minor axis of the elliptical core has to be accompanied by the gain in jet size on the major axis in order to satisfy the continuity condition in flow. In other words, the cross-sectional are in flow should be constant. Although the two dimensional nature of the image data does not distinguish between an elliptical cross section and a circular one, occasional observation of a smaller jet thickness at 15~m/s with 10~T field as opposed to a 5~T indicates that the jet cross section might vary between the major and minor axis of an elliptical core. It is important to note that within the axial distance of interest, the jet diameter is approximately constant. Therefore, references to "larger jet height" should be interpreted to mean larger distortions of the jet cross section.
Since the jet and solenoid field are cylindrically symmetric, it is hard to estimate in what direction
the jet is going to be distorted but the ratio of the deflection can be determined experimentally.
The ratio also can be compared with the transverse magnetic pressure $B^{2}/2\mu$ considering the
reversed direction of deflection on each plot.
Samulyak \cite{Samulyak2007} suggested that the deflection ratio of jet size $\triangle R / R_{o} $ is proportional
to the magnitude $ \textbf{B}_{o}^{2} / U$, where the governing MHD equations
and free jet boundary condition including Maxwell's equations using low magnetic Reynolds approximation
are employed and the Hg jet deflection in magnetic field is calculated by using a hybrid of Eulerian and
Lagrangian method, so called Front tracking method.
Figure~\ref{fig:jet_height_comparison}(a) shows the deflection ratio of Hg jet along the distance from nozzle
at 10~T and 15~T magnetic field \cite{Samulyak20072}.
The magnetic field stabilizes the Hg jet surface so that the jet surface is
getting flattened. In MHD simulation, constant 1~cm diameter of Hg jet is considered. Although the
magnetic field causes the jet surface flattening, the nature of turbulence such as growth of jet size is
observed in experiment. Therefore, in order to avoid such a turbulent nature between simulation and
experiment, the ratio of jet deflection ratio between 10 T and 15 T is evaluated to see the comparison
of the magnetic field effect $ \textbf{B}_{o}^{2} / U$ between Fig.~\ref{fig:jet_height_comparison}(a) and
Fig.~\ref{fig:jet_height_reduced}, which is shown in Fig.~\ref{fig:jet_height_comparison}(b).
It shows somewhat consistency at upstream, but still the ratio diverges as the jet flows to downstream.

As expected, jet motion in a magnetic field behaves differently, depending on the angle between the axis of magnet and the axis of jet, as a result of the differences in the magnitude of components of magnetic field \cite{Samulyak2006}.
Figure~\ref{fig:magnet_field}(a) and (b) show the axial and radial components of the
magnetic field in a solenoid.  Figure~\ref{fig:magnet_field}(c) and (d) show the transverse and longitudinal components of the magnetic field along the jet axis at different crossing angles. As the crossing angle increases, the transverse component of the magnetic field increases, but with no significant change in the longitudinal component of the magnetic field. An increase of the transverse component of the magnetic field raises the induced axial current on the Hg jet. Therefore, the angle of the Hg jet is launched at 33~milliradian with respect to the axis of solenoid magnet.

The jet surface can readily be extracted from each collected image.
The jet axis is approximated by fitting the averaged positions between top surface and bottom surface.
This jet axis is moved with an offset until it interferes the top surface bottom surface.
The amount of fluctuations of surface is measured by getting the difference between the
fluctuation surfaces and the interfering jet axis on a RMS scale.
Let $\delta(r,t)$ denotes the probability of turbulence at r, such that $\delta$ is defined as 0 in
the non-turbulent fluid, where the background is considered here, and is defined as 1 in the
turbulent fluid, where the jet is considered here. Time average of $\delta$ yields $\zeta(r)$,
the intermittency factor at r. The turbulent fluctuations are produced by the intermittency effect and
these fluctuations are significant for scalar quantities. The intermittency characteristics of the
turbulence are the appropriate input to be used in defining rough surface for a scattering analysis.
When the intermittency phenomenon is present, the conventional turbulent fluctuation is modified
by the intermittency function and there is an additional contribution depending on the difference
between the mean turbulent quantity and the non-turbulent quantity \cite{Yen}.
However, the probability of the fluctuating jet surface area is introduced to define the intermittency
in the following work.
The pixel information along the jet axis by changing the translational offset is summed up to
represent the intermittency of jet on the top and bottom surface. The intermittency within the jet
represents 1 and it is gradually decrease to 0 at the background. The intermittency is between 0 and i
at the jet surface depending on the surface fluctuations. Figure~\ref{fig:intermit} shows the intermittency
as a function of magnetic field and time. Total evaluated time is 160~$\mu$s.
Without magnetic field, the slope of intermittency at the jet surface is broad and it is oscillating
as a function of time. With higher magnetic field, the slope of intermittency at the jet surface is
more steep and it keeps same shape with respect to time. This result clearly tells that the magnetic
field suppresses the fluctuation of jet surface.

Figure~\ref{fig:disturb} shows the measured fluctuations on the jet surface.
Surface fluctuations is monotonically decreasing and the surface is flattened approximately
up to 5~T. The fluctuations at Viewport 3 (downstream) is larger than that at Viewport 1 (upstream) since
the tendency to be turbulent grows. The amount of fluctuations at top surface and bottom surface
of jet is almost same, though the magnetic field is varied. Thus, the symmetry on the jet surface in
terms of the surface variations such as fluctuations and wave amplitude is valid.
The amount of difference of surface fluctuations at Viewport 1 and Viewport 3 becomes same.
It indicates that the jet surface becomes flattened at $\sim$ 5~T in flow velocity 15~m/s.
The decreased amount of surface fluctuation at Viewport 1 and Viewport 3 is $\sim$ 0.5 and 1.5~mm
RMS respectively. This explains why the jet height is reducing from 0~T to 5~T in Fig.~\ref{fig:jet_height_reduced}. The magnetic field makes the wavelength on the jet surface
increases. Correspondingly, the wave propagation speed is increasing. Thus, it causes $\textrm{Re}_{cr}$
to increase and the flow becomes laminar due to the stabilization by the magnetic field.
The transverse component of magnetic field prevails more over the jet stabilization.
Though there is some measurement errors due to the saturation in image brightness,
the measurement could show the field effect to the reduction of fluctuation on jet surfaces.

The these observations are supported by previous results. For example, several investigations have suggested that magnetic field suppresses turbulent fluctuations in conducting liquid by stabilizing the flow
\cite{Shercliff1956, Gold, Kozyrev, Bernshtam}, where stabilization is judged by an increase in the
characteristic wavelength of the flow.

\subsection{Trajectory of mercury jet projectile in a magnetic field}

The Hg jet and the beam are launched at 33 and 67~milliradian with respect to the magnetic axis
respectively. The trajectory of Hg jet projectile is acted upon by gravity, which is represented as follow:

\begin{equation}
 \label{e:eq41c}
\begin{split}
&t = \frac{x}{v_{o} \cos \theta} \\
&y = - \frac{g}{2} \  t^{2} + v_{o} \sin \theta \ t + y_{nozzle}  \\
&| \textit{v} |  = \sqrt{\textit{v}_{o}^{2} - 2 g x \tan \theta + (\frac{g x}{\textit{v}_{o} \cos \theta})^{2}  } \ ,
\end{split}
\end{equation}

where $x$ is the jet traveling distance, $y$ is the height of jet centroid at $x$, $y_{nozzle}$ is the
vertical position of nozzle, $\textit{v}_{o}$ is the launched velocity, and $\theta$ is the launched angle
of Hg jet. Based on the governing trajectory equation  Eqn.~(\ref{e:eq41c}), fit function of the jet flow height can be expressed as

\begin{equation}
 \label{e:eq51d}
y = a_{1} + b_{1} x - \frac{g (1 + b_{1}^{2}) x^{2}}{2 c_{1}^{2}} ,
\end{equation}

where $a_{1} = y_{nozzle}$, $b_{1} = \tan \theta$, and $c_{1} = \textit{v}_{o}$. The values and error are given in Table~\ref{tab:fitdata_ch4}.
The distance of jet elevation is determined by measuring the distance from the magnetic axis at center of
each window to the jet axis, which is approximated by fitting the averaged positions between top surface
and bottom surface.
Figure~\ref{fig:position_jet_beam} shows the trajectory of Hg jet and it's effect by the magnetic
field and gravity. The solid line represents the globally fitted value using the trajectory of projectile with different initial launching speed of jet for the case of 15~m/s and 20~m/s respectively. It shows that the trajectory of Hg flow approximately agrees well with the trajectory of projectile for both 15~m/s and 20~m/s shots. Experiment shows that the trajectory of the Hg jet is parabolic. The magnetic field
caused some elevation of Hg jet closer to the center of magnetic field. As the jet moves to downstream,
magnetic field effect is more clearly observed since the jet is more likely to elongate to the axial direction.
At 15~T, the elevation of jet is observed from Viewport 1 to Viewport 4.
It shows that the magnetic force is overcoming the inertia force at 15~T similarly as there is the
increase in jet height at 15~T. The overall increase of the jet elevation in upstream,
midstream, and downstream at 15~T may have been caused by the asymmetric change of jet height.
Possibly the stable equilibrium between magnetic force and gravitational force could be varying
according to the variation of magnetic field \cite{Geim}.

The beam trajectory is also given to show the overlap with the Hg jet.
It is shown that the overlap length is $\sim$ 30~cm when we consider the height of jet at various
position with various magnetic field.

Based on the result of the jet trajectory, the angle of jet axis at Viewport 2 (midstream) is
determined by the trigonometric approach using the elevation of jet and the distance along the
magnetic axis between Viewport 1 and Viewport 3. Figure~\ref{fig:jet_angle_vp2} shows the estimation of
jet angle at centner of magnetic axis (Viewport 2), which is  approximately 7 $\sim$ 11~milliradian.
The jet angle is slightly decreasing with higher magnetic field, which indicates that the jet is
more likely to move horizontally following the field line direction.

\section{Dynamics of Liquid Jet Flow from Nozzle}

\subsection{Jet flow in surrounding medium}

Lee \cite{Lee} investigated the phenomenon of air wake caused by a cylindrical jet emerging from a nozzle and
showed the boundary layer of jet by applying continuity of jet mass and matching the loss of jet momentum with
air drag on the jet. Figure~\ref{fig:jetboundary_nozzle} depicts schematically the boundary layer of jet emerging from a nozzle.
The conservation of axial momentum and the rate of momentum loss to the skin friction on the jet
and the continuity of the liquid jet are expressed as follows:

\begin{equation}
 \label{e:eq414}
2 \pi \rho_{a} \int_{0}^{\delta(x)} [a(x)+y] v^{2}(x,y) dy + \rho_{l} \pi a^{2}(x) v_{l}^{2} (x) = \rho_{l} \pi a_{o}^{2} v_{lo}^{2} ,
\end{equation}

\begin{equation}
 \label{e:eq415}
\frac{d}{dz} [\pi \rho_{l} a^{2} (x) v_{l}^{2} (x) ] = 2 \pi a(x) \mu_{a} \frac{\partial v}{\partial y} |_{y=0} ,
\end{equation}

and

\begin{equation}
 \label{e:eq416}
\pi a^{2} (x) v_{l} (x) = \pi a_{o}^{2} v_{lo} ,
\end{equation}

where velocity, density, and viscosity are denoted by $v$, $\rho$, and $\mu$, respectively, with subscripts $a$ and $l$ for air and liquid, respectively. The subscripts $o$ denotes the initial values at the nozzle. For boundary layer analysis of cylindrical objects, because of the diverging flux characteristics in
radial direction, a logarithmic profile is the most appropriate \cite{Stewartson, Glanert}.
Thus, the velocity profile of the air induced by the liquid jet is assumed to be as follow:

 \begin{equation}
 \label{e:eq417}
v(x,y) = v_{l}(x) \{ 1 - \frac{1}{\beta (x)} \ln [ 1 + \frac{y}{a(x)} ] \}
\end{equation}

and

 \begin{equation}
 \label{e:eq418}
\frac{\delta(x)}{a(x)} = e^{\beta(x)} - 1 .
\end{equation}

Substitution of Eqns.~(\ref{e:eq417}) into Eqns.~(\ref{e:eq414}) $\sim$ Eqns.~(\ref{e:eq416}) results in

 \begin{equation}
 \label{e:eq419}
\bar{v}_{l}(\bar{x}) = \frac{v_{l} (\bar{x})}{v_{lo}} = \frac{1}{1 - \bar{\rho} [ 1 + \frac{1}{\beta} - \frac{1}{2 \beta^{2}}(e^{2 \beta} - 1) ]}
\end{equation}

and

 \begin{equation}
 \label{e:eq420}
\frac{d \beta(\bar{x})}{d \bar{x}} = \frac{\beta^{2} - \bar{\rho} [ \beta^{2} + \beta - \frac{1}{2} (e^{2 \beta} -1)]}{\beta (1 + e^{2 \beta}) - (e^{2 \beta} - 1)} ,
\end{equation}

where $ \bar{x} = \frac{4 x}{\textrm{Re}_{e} a_{o}}$, $\textrm{Re}_{e} = \frac{2 a_{o} \rho_{a} v_{lo}}{\mu_{a}}$, and $\bar{\rho} = \frac{\rho_{a}}{\rho_{l}}$.
For a given value of $\rho$, the stream velocity $\bar{v}_{l}(\bar{x})$ and boundary layer thickness $\bar{\delta} (x)$ are obtained.
As seen in Fig.~\ref{fig:flowvel_simulation}, the Reynolds number plays its role implicitly and
this makes the density ratio $\bar{\rho}$ to be varied. Since the cylindrical jet has larger volumes,
for the initial momentum of the jet to be maintained, the liquid density must be reduced and the value
of the density parameter to be used must be modified to $\bar{\rho} = \rho_{a} D^{2} / (\rho_{l} d_{o}^{2})$, where $D$ and $d_{o}$ denote diameter of jet and nozzle, respectively.

\subsection{Pressure loss and magnetic effect on the Hg delivery pipe}

Fig.~\ref{fig:hgpressure} (a) and (b) show the pipe inlet pressure for
driving jet in various magnetic field strengths.
The Hg jet is driven by the piston in syringe and the piston velocity is measured by position sensor.
The piston velocity determines the flow rate so that the dynamic pressure head at pipe inlet is
determined using the conservation law of flow rate. The pressure sensor installed at the pipe wall measures
the static pressure. No significant pressure drop is observed at the pipe inlet in magnetic field strengths. It indicates that the driving pressure in pipe for nozzle is at same conditions
regardless of the magnetic field variation.

To obtain the jet velocity, the distance traveled by a fixed point on the jet surface is
tracked over a given time period. Figure~\ref{fig:jet_velocity} (a) shows the jet velocity measured
at Viewport 1, Viewport 2, Viewport 3, and Viewport 4 in various magnetic field strengths.
Note that this velocity does not significantly change with the imposition of a magnetic field. Therefore, considering
the measurement error in Fig.~\ref{fig:jet_velocity} (a), the averaged flow velocity, regardless of
magnetic fields, can reasonably indicate the flow velocity given in Fig.~\ref{fig:jet_velocity} (b).
This explains why the pressure is approximately constant in the pipe, consistent with the
report \cite{Graves20071}.

Another interesting result is that the cross section of Hg jet is more likely
to be elliptical since the longitudinal jet flow velocity is constant from upstream to downstream.
Regardless of the magnetic field, the Hg jet does not show jet velocity change. Thus, the jet is changing
its shape once it leaves the nozzle from circular to elliptical.
Hence, the result in Fig.~\ref{fig:jet_height_reduced} should be again interpreted by the result in
Fig.~\ref{fig:jet_velocity} in the manner that the jet height at 5 T is elongated on the minor axis
followed by the reduction of jet height on the major axis of the elliptical core, and the jet is
deflecting further at 10 T. However, the jet height at 15 T is elongated on the major axis, which is
manifested by the comparison between the ratio of the reduction of jet height and the increased ratio
of the jet height at 15 T. This approach is already mentioned in the above, but it is examined again.

Considering that the driving pressure and the jet velocity are not significantly
changed in various magnetic fields, it is concluded that the longitudinal magnetic field does not affect
to the pressure loss or velocity degradation while Hg passes inside of the solenoid magnet two times along with
the direction of magnetic field line. It is reported that the gradient of longitudinal jet velocity
depends on the integration of gradient of longitudinal magnetic field along the magnetic axis plus
it's multiplication to longitudinal magnetic field itself \cite{Gallardo2002}. It is expressed as follow:

\begin{equation}
 \label{e:eq41b}
\Delta \textit{v} (x) = \frac{\kappa}{\rho} \frac{r_{o}^{2}}{8} ( \ \int_{x_{1}}^{x_{2}} (\frac{d B_{x}}{d x})^{2}  + \frac{d}{d x} (B_{x} \frac{d B_{x}}{d x}) \ d x \ ) \ ,
\end{equation}

where $r_{o}$ is the radius of jet and $\kappa$ is electrical conductivity.
Since the gradient of magnetic field is increasing (plus) at entrance and decreasing (minus) at exit,
it seems that there is an increasing velocity gradient (acceleration) at upstream and
decreasing velocity gradient (deceleration) at downstream but it is $\leq$ 0.5~m/s due to
the relatively high density comparing with the electrical conductivity only if we consider the effect by
the magnetic field. The experimental result shows slight effect of magnetic field.

\subsubsection{\textit{pressure loss in pipe flow}}

Hg delivery pipe is connected from syringe pump to nozzle and it is passing parallel with solenoid
magnetic field line next to the primary containment.
A loss coefficient is defined as follows:

\begin{equation}
\begin{split}
 \label{e:eq43b}
&(h_{loss}) = K_{1} \frac{\textit{v}_{1}^{2}}{2 g} + K_{2} \frac{\textit{v}_{2}^{2}}{2 g} + \ldots + K_{N} \frac{\textit{v}_{N}^{2}}{2 g} \ , \\
&A_{1} \textit{v}_{1} = A_{2} \textit{v}_{2} = \ldots = A_{N} \textit{v}_{N} = A_{R} \textit{v}_{R} \ ,
\end{split}
\end{equation}

where the subscript $R$ signifies a reference location and $K$ represents the loss coefficient.
The general thermodynamic loss, so called the head loss $h_{loss}$, is defined as follow:

\begin{equation}
 \label{e:eq44b}
\int_{1}^{2} \delta F = (h_{loss})_{1,2} = \frac{p_{1} - p_{2}}{\rho g} + \frac{\textit{v}_{1}^{2} - \textit{v}_{2}^{2}}{2 g} + (z_{1} - z_{2}) \ .
\end{equation}

Darcy-Weisbach equation is given to express the head loss of wherever the density is constant as follow:

\begin{equation}
 \label{e:eq42b}
\frac{\Delta p}{\rho g} = f \frac{L}{d} \frac{U^{2}}{2 g} \ ,
\end{equation}

where f, L, d are friction factor, the pipe length, and the diameter of pipe respectively.
Considering that the \textrm{Re} = 1800000 and e/d = 0.002
for commercial steel in terms of Nikuradse's sand grain scale, turbulent friction factor f via Moody plot
or by Colebrook Eqn.~(\ref{e:eq44c}) is approximated to 0.024.
Colebrook simply combined the expressions for the friction factor for smooth and rough pipes into a
single transition equation of the equivalent form as follow:

\begin{equation}
 \label{e:eq44c}
\frac{1}{\sqrt{f}} = 1.74 - 2 \ log \ ( \ 2 \frac{e}{d} + \frac{18.7}{\textrm{Re} \sqrt{f}} \ ) \ .
\end{equation}

Note that Colebrook's expressions for the friction factor in the transition region reduces to Prandtl's
smooth pipe equations when the relative roughness approaches zero, and reduce to von Karman's fully rough
pipe equation at very high pipe Reynolds number \cite{Benedict}.

The loss coefficients for elbows are presented as follows, where a and R represent the inside radius
of the elbow and the radius of curvature of the centerline of the elbow respectively.
For $\textrm{Re} (a/R)^{2} > 91$, the loss coefficient is expressed as follow \cite{Ito}:

\begin{equation}
 \label{e:eq45b}
K_{elbow} = 0.00241 \ \alpha \ \theta \ ( \ \frac{R}{a} \ )^{0.84} \textrm{Re}^{-0.17} \ ,
\end{equation}

where $\theta$ is the bend angle in degrees and $\alpha$ is an empirical factor given as:

\begin{equation}
 \label{e:eq46b}
\alpha_{\theta = 90^{\circ}} = 0.95 + 17.2 \ ( \ \frac{R}{a} \ )^{-1.96} \ .
\end{equation}

Inputting $R$=1.942 and $a$=0.442, $\alpha$ = 1.9 and $K_{elbow}$ = 0.1232.
A correction term is applied to the 90$^{\circ}$ elbow to determine the loss coefficient
for arbitrary angle of elbow \cite{SAE}.

\begin{equation}
 \label{e:eq47b}
K_{\theta} = ( \ C_{\theta} \ )_{elbow} \ K_{90^{\circ}} \ ,
\end{equation}

where $C_{elbow}$ is given in the referenced manual \cite{SAE}. The $C_{\theta}$ is 0.28 at
$\theta$ = 23$^{\circ}$ and $K_{\theta}$ = 0.0345.

The loss coefficient for the reducer or well-rounded inlet loss is
$K_{reducer}$ = 0.05 based on the flow area of the smaller piping section \cite{Benedict}.

Finally, the loss coefficient for the abrupt contraction is given based on the velocity at exit
as follow \cite{Benedict}:

\begin{equation}
 \label{e:eq49b}
 \begin{split}
&K_{contraction} = (\frac{1}{C_{D}^{2}} - 1 ) (1 - \beta^{4}) \ , \\
&C_{D} = \frac{Q_{acutal}}{Q_{ideal}} \ ,
\end{split}
\end{equation}

where the discharge coefficient $C_{D}$ is given in reference \cite{Benedict}.
The mean discharge coefficient is given as 0.815 based on the water tests in short pipes.
According to Eqn.~(\ref{e:eq49b}), this yields a maximum loss coefficient at $\beta = 0$ of 0.506.
Assuming $\beta = A_{2} /A_{1} = 0.9$, $K_{contraction}$ yields 0.1738.

The head losses and the contribution of each geometry are given in Table~\ref{tab:headloss}.
Total length of pipe is 87.1~inch. The diameter of inside pipe is 0.884~inch. The diameter of
inside nozzle is 0.4~inch. Total pressure head loss is 2.3358~m., which corresponds to $\sim$ 16~$\%$
of input pressure head. The main loss is caused by large length with friction by surface roughness inside
pipe, which is over $\sim $ 50~$\%$. The loss from pipe bend is somewhat low comparing with others.

Based on the calculated head loss, the jet velocity at nozzle is determined assuming the pressure
right after nozzle exit is atmospheric. The pipe inlet pressure is given in Fig.~\ref{fig:hgpressure}(a) and (b). The elevation of the pipe inlet and the nozzle is $\sim$ 2.9~inch.
The calculated jet velocity from nozzle including the pressure loss in pipe is 15 $\pm$ 1~m/s, which is
$\sim $ 10 $\%$ larger than the measured result in Fig.~\ref{fig:jet_velocity}, where the jet velocity is
$\sim$ 13.5~m/s. Jet velocity in nozzle position was reported in Fig.~\ref{fig:hgpressure}(c) with $\sim$
15.8~m/s by using incompressible constant flow rate between pipe inlet and nozzle exit, where piston size is
10 inch, nozzle inside size is 0.4 inch, and moving position of piston for injection of Hg jet was
measured with respect to 10~Hz of data acquisition rate via NI hardware \cite{Graves20072}.
The magnetic field increases the fluid pressure by an amount of $B^{2}/2 \mu$, in directions perpendicular
to the magnetic field, and decreases the fluid pressure by the same amount, in the parallel direction of the magnetic field. The fluid pressure including the magnetic pressure has to balance with the atmospheric pressure
and surface tension of jet, and it should satisfy the continuity condition.
The fluid pressure will find equilibrium point since the fluid pressure perpendicular to the magnetic
field line is mutually symmetric. Therefore, the jet is changing to be elliptical.
Hence, the pressure drop is not occurred significantly and correspondingly the longitudinal jet velocity is not changed with magnetic fields in Fig.~\ref{fig:jet_velocity}.

\subsubsection{\textit{measurement of wall tap pressure}}

Wall tap is used in order to measure static pressure, wherein small pressure tap is located at a
point on pipe surface so that it does not disturb the fluid.
Tap size error arises because of a local disturbances of the boundary layer.
At tap $\textrm{Re}$ greater than 385, the error in static pressure caused by the tap size is
given as follow:

\begin{equation}
 \label{e:eq411b}
\frac{e_{tap}}{\tau} = 0.269 \ (\textrm{Re}_{d}^{*})^{0.353} \ ,
\end{equation}
where $\frac{e_{tap}}{\tau} = 12.74$ and

\begin{equation}
 \label{e:eq410b}
\begin{split}
&\textrm{Re}_{d}^{*} = \frac{\textit{v}^{*} d_{tap}}{\nu}   \\
&\textrm{Re}_{d}^{*} = \sqrt{\frac{f}{8}} \ (\frac{d_{tap}}{d}) \ \textrm{Re} \ ,
\end{split}
\end{equation}

where $d_{tap}$ is the tap diameter, $\textrm{Re}_{d}^{*}$ is the tap $\textrm{Re}$ number,
and $\textit{v}^{*}$ is the friction velocity. The friction factor is 0.024. The inside diameter of tap and
inside diameter of pipe are 0.5, 0.884 inch respectively, which yields $\textrm{Re}_{d}^{*} = 55764$.
Combining the Darcy friction factor with the wall shear stress yields

\begin{equation}
 \label{e:eq412b}
f = 4 \ (\frac{\tau}{\rho \textit{v}^{2} / 2 g}) \ .
\end{equation}

Therefore, the error in a static pressure can be expressed in a non-dimensionalized form
by the dynamic pressure $p_{dynamic}$ as follow:

\begin{equation}
 \label{e:eq413b}
\frac{e_{tap}}{p_{dynamic}} = (\frac{e_{tap}}{\tau}) \frac{f}{4} \ ,
\end{equation}

where $\frac{e_{tap}}{p_{dynamic}}$  = 0.0764. The error of static pressure
in Fig.~\ref{fig:hgpressure} (a) is estimated to give 7.64~$\%$ uncertainty of the
dynamic pressure in Fig.~\ref{fig:hgpressure} (b).

\setlength{\baselineskip}{12pt}


\newpage
\clearpage

\begin{landscape}
\begin{table}
\centering
\tablecaption{\label{tab:fiderror}Error estimation of fiducial length at each Viewport.}
\begin{supertabular}{|l|l|l|}
\hline
\hline
Viewport number \ \ \ \ \ \ \ \ \ \ \ \ \ \ & Fiducial length (cm) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ &  Scaling factor \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\
\hline
1 & 1.0 $\pm$ 0.095 & 1.0 (reference) \\
2 & 1.0 $\pm$ 0.091 & 4.3 $\pm$ 0.81  \\
3 & 1.0 $\pm$ 0.062 & 1.0 $\pm$ 0.16  \\
4 & 1.0 $\pm$ 0.067 & 4.3 $\pm$ 0.70 \\
\hline
\hline
\end{supertabular}
\end{table}
%\end{landscape}

%\begin{landscape}
%\centering
%\caption{\label{tab:headloss}Pressure head losses by geometry in pipe for mercury loop.}
\begin{table}
\centering
\tablecaption{\label{tab:headloss}Estimation of pressure head losses by geometry of pipe in Hg loop.}
\begin{supertabular}{|l|l|l|}
\hline
\hline
Geometry in pipe for Hg loop & Calculated pressure head loss &  Percentage in total pressure head loss ($\%$)  \\
\hline
Friction by surface roughness & 1.4176 & 60.7 \\
Elbows in pipe bend (  3 $\times$ 90$^{\circ}$, 2 $\times$ 23$^{\circ}$  ) & 0.2629 & 11.3  \\
Reducer, Contraction in nozzle & 0.6553 & 28  \\
\hline
\hline
\end{supertabular}
\end{table}
\end{landscape}

\newpage
\clearpage

\begin{landscape}
\begin{table}[!hbt]
\centering
\tablecaption{\label{tab:fitdata_ch4} Parameterized coefficients, its error, and statistics summary of fit function in figures.}

\begin{supertabular}{|l|l|l|l|l|l|l|l|l|l|l|}
\hline
\hline
 $\textbf{Figure}$ & $\textbf{1}$ & $\textbf{2}$ & $\textbf{3}$ & $\textbf{4}$ & $\textbf{5}$ & $\textbf{6}$ & $\textbf{7}$ & $\textbf{8}$ & $\textbf{9}$  & $\textbf{10}$ \\
\hline
~\ref{fig:position_jet_beam}(B=0~T,V=15~m/s)  & -0.01448  & 9.97E-04 &  0.03375  & 0.00379 & - & - & - & - &  13.6445 &  0.85213 \\
~\ref{fig:position_jet_beam}(B=5~T,V=15~m/s)  & -0.01448 & 9.97E-04 & 0.03375 & 0.00379 & - & - & - & - & 13.85258 & 0.89937
 \\
~\ref{fig:position_jet_beam}(B=10~T,V=15~m/s)  & -0.01448 & 9.97E-04 & 0.03375 & 0.00379 & - & - & - & - & 14.13407 &  0.96089
 \\
~\ref{fig:position_jet_beam}(B=15~T,V=15~m/s)  & -0.01448 & 9.97E-04 & 0.03375 & 0.00379 & - & - & - & - & 14.48514 & 0.99102
  \\
~\ref{fig:position_jet_beam}(B=15~T,V=20~m/s)  & -0.01448 & 9.97E-04 & 0.03375 & 0.00379 & - & - & - & - & 18.85852 & 2.2851
 \\
\hline
\hline
 $\textbf{Figure}$ & $\textbf{11}$ & $\textbf{12}$ & $\textbf{13}$ & $\textbf{14}$ & $\textbf{15}$ & $\textbf{16}$ & $\textbf{17}$ & $\textbf{18}$ & $\textbf{19}$ &  \\
\hline
~\ref{fig:position_jet_beam}(B=0~T,V=15~m/s)  & - & - & - & - & 20 & 13 & 25.15504 & 0.92629 & 0  & \\
~\ref{fig:position_jet_beam}(B=5~T,V=15~m/s)  & - & - & - & - & 20 & 13 & 25.15504 & 0.92629 & 0  & \\
~\ref{fig:position_jet_beam}(B=10~T,V=15~m/s)  & - & - & - & - & 20 & 13 & 25.15504 & 0.92629 & 0  & \\
~\ref{fig:position_jet_beam}(B=15~T,V=15~m/s)  & - & - & - & - & 20 & 13 & 25.15504 & 0.92629 & 0  & \\
~\ref{fig:position_jet_beam}(B=15~T,V=20~m/s)  & - & - & - & - & 20 & 13 & 25.15504 & 0.92629 & 0  & \\
\hline
\hline
\end{supertabular} \\

\ \\
%\normalsize
\begin{flushleft}

\shortstack[l]{\vrule width 8.5in height 0.4pt\\
        \ \ $\textbf{1}$ : $a_{1}$ value, $\textbf{2}$ : $a_{1}$ standard deviation,   \\
       \ \\
        \ \ $\textbf{3}$ : $b_{1}$ value, $\textbf{4}$ : $b_{1}$ standard deviation, $\textbf{5}$ : $b_{2}$ value, $\textbf{6}$ : $b_{2}$ standard deviation, \\
       \ \\
        \ \ $\textbf{7}$ : $b_{3}$ value, $\textbf{8}$ : $b_{3}$ standard deviation , $\textbf{9}$ : $c_{1}$ value, $\textbf{10}$ : $c_{1}$ standard deviation,
         \ \\
         \ \ $\textbf{11}$ : $c_{2}$ value, $\textbf{12}$ : $c_{2}$ standard deviation, $\textbf{13}$ : $c_{3}$ value, $\textbf{14}$ : $c_{3}$ standard deviation, \\
         \ \\
        \ \ $\textbf{15}$ : Number of points, $\textbf{16}$ : Degrees of freedom, $\textbf{17}$ : Reduced $\chi^{2}$, $\textbf{18}$ : Adjusted $\textbf{R}^{2}$, $\textbf{19}$ : $\chi^{2}$ probability.  }
       \ \\
\end{flushleft}

\end{table}
\end{landscape}


\newpage
\clearpage
\begin{landscape}
\begin{figure}
\centering
\includegraphics{jet15ms2}
\caption{Hg jet flows as observed from each of 3 Viewports. The jet flows from left to right on each image. The first, second, and third columns represent Viewport 1, 2, and 3, respectively. The individual caption shows the applied magnetic field. The jet velocity is 15~m/s. Images on Viewport 2 has a 14$^{\circ}$ clockwise rotation due to the SMD software. a.) B=0~T. b.) B=0~T. c.) B=0~T. d.) B=5~T. e.) B=5~T. f.) B=5~T. g.) B=10~T. h.) B=10~T. i.) B=10~T. j.) B=15~T. k.) B=15~T. l.) B=15~T. }
\label{fig:jet15ms}
\end{figure}
\end{landscape}
\clearpage

\newpage
\clearpage
\begin{landscape}
\begin{figure}
\centering
\includegraphics{jet20ms2}
\caption{Same as Fig.~\ref{fig:jet15ms} but with a jet velocity of 20~m/s. a.) B=0~T. b.) B=0~T. c.) B=0~T. d.) B=5~T. e.) B=5~T. f.) B=5~T. g.) B=10~T. h.) B=10~T. i.) B=10~T. j.) B=15~T. k.) B=15~T. l.) B=15~T. }
\label{fig:jet20ms}
\end{figure}
\end{landscape}
\clearpage

\newpage
\clearpage
\begin{landscape}
\begin{figure}
\centering
\includegraphics{image_convert}
\caption{Image data conversion for image analysis. a.) Collected image data.  b.) 2 bit scaled image data. }
\label{fig:image_convert}
\end{figure}
\end{landscape}
\clearpage

\newpage
\clearpage
\begin{landscape}
\begin{figure}
\centering
\includegraphics{data_example3}
\caption{Jet height determination from image analysis. a.) Sensitivity of threshold in a 2 bit scaled image conversion. b.) Histogram of number of events in the jet height measurement. }
\label{fig:data_example}
\end{figure}
\end{landscape}
\clearpage

\newpage
\clearpage
\begin{landscape}
\begin{figure}
\centering
\includegraphics{magnet_Bfield3}
\caption{Calculated solenoid magnetic field map. a.) Radial field map. b.) Axial field map. c.) Transverse component of magnetic field along jet axis. d.) Longitudinal component of magnetic field along jet axis. } \label{fig:magnet_field}
\end{figure}
\end{landscape}
\clearpage

\newpage
\clearpage
\begin{landscape}
\begin{figure}
\centering
\includegraphics[width=8in]{jet_height7}
\caption{Hg jet height measurement from direct averaging of vertical height in magnetic fields on each image. } \label{fig:jet_height_reduced}
\end{figure}
\end{landscape}
\clearpage

\newpage
\clearpage
\begin{landscape}
\begin{figure}
\centering
\includegraphics{jet_height_comparison}
\caption{Comparison of Hg jet deflection ratio at 15~T to that at 10~T. a.) Numerical calculation of
deflection ratio \cite{Samulyak20072}. b.) Comparison of jet deflection ratio. } \label{fig:jet_height_comparison}
\end{figure}
\end{landscape}
\clearpage

\newpage
\clearpage
\begin{landscape}
\begin{figure}
\centering
\includegraphics{intermit1}
\caption{Intermittency of Hg jet at Viewport 2. The jet velocity is 15~m/s. a.) B=0~T. b.) B=5~T. c.) B=10~T. d.) B=15~T (continued).} \label{fig:intermit}
\end{figure}
\end{landscape}
\clearpage

\newpage
\clearpage
\addtocounter{figure}{-1}

\begin{landscape}
%\addtocounter{subfigure}{1}
\begin{figure}
\centering
\includegraphics{intermit2}
\caption{Intermittency of Hg jet at Viewport 2. The jet velocity is 20~m/s. e.) B=0~T. f.) B=5~T. g.) B=10~T. h.) B=15~T.} \label{fig:intermit}
\end{figure}
\end{landscape}
\clearpage

\newpage
\clearpage
\begin{landscape}
\begin{figure}
\centering
\includegraphics[width=8in]{jetdisturbance2}
\caption{Measurements of surface fluctuations of Hg jet at upstream and downstream in magnetic fields.}
\label{fig:disturb}
\end{figure}
\end{landscape}
\clearpage

\newpage
\clearpage
\begin{landscape}
\begin{figure}
\centering
\includegraphics[width=8in]{position_jet_beam5}
\caption{Trajectory of beam axis and Hg jet axis with respect to magnetic axis in magnetic fields.
Solid line represents the globally fitted values using the formula of trajectory of projectile with different launching velocity.} \label{fig:position_jet_beam}
\end{figure}
\end{landscape}
\clearpage

\newpage
\clearpage
\begin{landscape}
\begin{figure}
\centering
\includegraphics[width=8in]{jet_angle_vp21}
\caption{The estimation of angle of Hg jet axis at the center of Viewport 2 as a function of magnetic field.} \label{fig:jet_angle_vp2}
\end{figure}
\end{landscape}
\clearpage

\newpage
\clearpage
\begin{landscape}
\begin{figure}
\centering
\includegraphics[width=7in]{jetboundary_nozzle}
\caption{Boundary layer induced by a jet emerging from a nozzle.} \label{fig:jetboundary_nozzle}
\end{figure}
\end{landscape}
\clearpage

\newpage
\clearpage
\begin{landscape}
\begin{figure}
\centering
\includegraphics[width=8in]{flowvel_simulation}
\caption{Stream velocity and boundary layer thickness for various values of density ratio.} \label{fig:flowvel_simulation}
\end{figure}
\end{landscape}
\clearpage

\newpage
\clearpage
\begin{landscape}
\begin{figure}
\centering
\includegraphics[width=7in]{hgpressure4}
\caption{Pipe inlet pressure for driving Hg jet. a.) Static pressure. b.) Dynamic pressure. c.) Jet velocity in nozzle \cite{Graves20071}.} \label{fig:hgpressure}
\end{figure}
\end{landscape}
\clearpage

\newpage
\clearpage
\begin{landscape}
\begin{figure}
\centering
\includegraphics[width=8in]{jet_velocity3}
\caption{Longitudinal Hg jet velocity in magnetic fields. a.) Velocity at each Viewport dependent of
magnetic fields. b.) Averaged velocity at each Viewport independent of magnetic fields.} \label{fig:jet_velocity}
\end{figure}
\end{landscape}
\clearpage

