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<p class="art-type" id="articleinfo">Research Article</p>
<p class="art-title">
Hypersonic Boundary Layer Flow at a Stagnation 
Point with Applied Magnetic Field and Vectored 
Surface Mass Transfer</p>
<p class="art-author"><?php $authors="
Rama Subba Reddy Gorla<sup>1*</sup>, Mahesh Kumari<sup>2</sup>"; echo (stristr($authors,$coauthor))?str_replace($coauthor,"<a href='".$extpath."authors/".$courl."' target='_blank'>".$coauthor."</a>",$authors):$authors; ?></p>
<p class="art-affl"><sup>1</sup>Professor of Aerospace Engineering, Ph.D., Department of Aeronautics and Astronautics, Air Force Institute of Technology, Wright 
Patterson Air Force Base, USA<br/><sup>2</sup>Research Scientist (Retired), Ph.D., Department of Mathematics, Indian Institute of Science, India</p>
<p class="art-aff"><b>*Corresponding author: <?php $corresponding_author="
Rama Subba Reddy Gorla"; echo ($coauthor!="" && $coauthor==$corresponding_author)?"<a href='".$extpath."authors/".$courl."' target='_blank'>".$coauthor."</a>":$corresponding_author;?></b>, 
Professor of Aerospace Engineering, Ph.D.
Department of Aeronautics and 
Astronautics,
Air Force Institute of Technology, Wright 
Patterson Air Force Base,
Dayton, Ohio 45433,
USA, E-mail: <a href="mailto:Rama.Gorla@afit.edu">Rama.Gorla@afit.edu</a></p>
<p class="art-aff"><b>Received:</b> October 25, 2023 <b>Accepted:</b> November 20, 2023 <b>Published:</b> December 2, 2023</p>
<p class="art-aff"><b>Citation:</b>: Gorla RSR, Kumari M. Hypersonic 
Boundary Layer Flow at a Stagnation Point 
with Applied Magnetic Field and Vectored 
Surface Mass Transfer. <i>Int J Aeronaut 
Aerosp Eng</i>. 2023; 3(1): 86-93.
doi: <a href="https://doi.org/10.18689/ijae-1000112">10.18689/ijae-1000112</a></p>
<p class="art-aff"><b>Copyright:</b> &copy; 2023 The Author(s). This work 
is licensed under a Creative Commons 
Attribution 4.0 International License, which 
permits unrestricted use, distribution, and 
reproduction in any medium, provided the 
original work is properly cited.</p>
<p><a href="<?php echo $extpath;?><?php echo $jres['journal_link'];?>/ijae-1000112.pdf" class="btn btn-danger pull-right" target="_blank">Download PDF</a></p>
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<div class="articlecontent">
<p class="art-subhead" id="abstract">Abstract</p>
<p class="art-para">A boundary layer analysis is presented for hypersonic flow in the vicinity of a 
stagnation point region of a blunt body under isothermal and adiabatic boundary 
conditions. Consideration is given to variable properties of air. It has been shown that 
surface drag and heat transfer rates may be controlled by applying magnetic field and 
vectored surface mass transfer. The range of Mach numbers considered is 2.5 to 10. As 
the magnetic field strength M increases, friction factor and heat transfer rate (in the case 
of isothermal surface) or surface temperature (in the case of adiabatic surface) increase. 
Friction factor and surface temperature (in the case of adiabatic surface) can be reduced 
by applying vectored surface mass transfer. </p>
<p class="art-para"><b>Keywords:</b> Hypersonic; compressible flow; boundary layer; magnetic field; vectored 
surface mass transfer</p>
<p class="art-subhead" id="intro">Introduction</p>
<p class="art-para">Aerodynamic heating is the heating of a surface due to flow of air at high speed. The 
kinetic energy of the air will be converted to thermal energy within the boundary layer. 
Prediction of the heat transfer rate to the surface at high Mach number flows is crucial 
to the design of thermal protection system. Tauber and Mennes (1986) made engineering 
estimates of the aerodynamic heating to an aerospace plane. According to their 
estimates, aerodynamic heating during ascent (about 650 W/cm2) dominates the design 
of the vehicle.</p>
<p class="art-para"> 
At hypersonic speeds, air cannot be considered as a simple mixture of diatomic 
oxygen and nitrogen. At a Mach number 20, the air temperature behind a normal shock 
wave for a blunt body can reach 6500 K during re-entry. The air undergoes considerable 
change in composition. Dissociation of oxygen and nitrogen occurs. The air becomes 
electrically conductive at high temperatures and can interfere with radio wave transmission 
and reception. The magnetic field can alter the drag and heat transfer characteristics.</p>
<p class="art-para">Klopfer and Yee (1988) investigated viscous hypersonic flow on a blunt body. 
Hoffmann et al. (1991) discussed the difficulties in predicting heat transfer rates for high 
speed flows. Qu et al. (2017) presented an analysis for hypersonic heating. All these 
studies assumed a constant value of specific heat of air. Zhang et al. (2017) provided a 
turbulence model for aero thermal prediction. Liu and Cao (2017) studied the heat 

transfer in hypersonic boundary layer over a flat plate by 
taking into account of variable properties.</p>
<p class="art-para">The flow in the vicinity of a stagnation point is always 
laminar. The present analysis is valid for stagnation point 
region. It is expected that far away downstream from the 
stagnation point, the flow undergoes transition due to finite 
disturbances and finally reaches fully established turbulent 
condition. If the initial amplitude of the disturbance is small, 
Tollmien-Schlichting (TS) waves will be observed. The two 
dimensional waves then develop into growing three 
dimensional structures by secondary instability process. These 
develop nonlinearly into lambda (Λ) - shaped structures, 
known as lambda vortices and the fluid velocity changes 
rapidly. When amplitudes reach large values, there will be a 
rapid breakdown to short scaled structures known as spikes. 
This will be followed by the onset of random behavior and the 
eventual development of a turbulent flow. This is often through 
the growth of isolated patches of turbulence or spots from the 
regions of the spikes that merge as they travel downstream.</p>
<p class="art-para">The friction factor and heat transfer rate in turbulent flow 
will be higher than laminar flow conditions, reaching about 
50% more. The present proposed methodology of using 
magnetic forces and vectored surface mass transfer has the 
potential of reduction of friction factor and heat transfer and 
hence delay the transition to turbulence. Turbulent boundary 
layers can be relaminarized by creating favorable pressure 
gradient by adjusting the body contour. We make brief 
comments on relaminarization here.</p>
<p class="art-para">Wall-bounded turbulence remains one of the least 
understood phenomena in fluid mechanics, despite its 
significance in many engineering fields. For example, 
approximately 50% to 80% of the total energy expenditure of 
commercial airplanes and container ships is used to overcome 
the turbulent frictional drag. The turbulent motions generate 
large skin friction on the surface of the spacecraft, and 
dissipate significant amount of energy through the downward 
turbulent energy cascade. In contrast, a relaminarized 
turbulent boundary layer (TBL) generates much reduced wall 
shear stress as discussed by Mukund et al. (2006), which is of 
great interest to both scientists and engineers. TBLs are 
commonly relaminarized when imposed with a strong 
favorable pressure gradient as discussed by Narasimha and 
Sreenivasan (1979). The favorable pressure gradient 
accelerates the bulk flow forward, which stabilizes the TBL by 
reducing the turbulent production while increasing the 
turbulent dissipation. Consequently, the wall shear stress 
drops over certain developing distance. To reduce the skin 
friction of wall-bounded turbulent flows, Wang and Gharib 
(2021) recently proposed a dynamic free-slip boundary 
method, which can be easily implemented to achieve 
significant wall shear stress reduction (by more than 40%).</p>
<p class="art-para">The present work is undertaken in order to study the 
effect of applied magnetic field and vectored surface mass 
transfer in the case of hypersonic flow at a stagnation point. 
The properties of air are assumed to be temperature-dependent. Consideration is given to the application of 
transverse magnetic field in high speed aerodynamics where 
the air is ionized and changing the velocity and temperature 
fields as well as the drag and surface heat transfer rates. By 
applying magnetic field, one can control and reduce the drag 
and surface heat transfer rates. Such a mechanism is of 
extreme importance for flow over hypersonic vehicles. Rossow 
(1957) was the first one to analyze the effect of magnetic field 
in the case of incompressible boundary layer flow. The use of 
vectored surface mass transfer is another mechanism to 
reduce surface heat transfer rate and finds application in 
boundary layer control on aerodynamic vehicles, film and 
transpiration cooling of rocket engines, turbomachinery 
blades etc. Gorla (1977) studied the effect of vectored surface 
mass transfer in plane wall jet flows.</p>
<p class="art-para"><b>
Governing Equations</b></p>
<p class="art-para">We consider the laminar, hypersonic boundary layer flow 
of an electrically conducting viscous fluid at a stagnation 
point. An applied magnetic field B<sub>0</sub> is imposed on the 
boundary layer. Blunt bodies generate strong shock waves. 
There is entropy change across the shock wave. The flow 
before the shock and external flow to the boundary layer after 
the shock can be assumed isentropic. We will consider both 
isothermal and adiabatic boundary conditions for the 
stagnation region. We consider variable properties and 
include vectored surface mass transfer. The governing 
boundary layer equations may be written as:</p>
<div class="art-img" id="e001">
<img src="<?php echo $imgpath;?>images/ijae-112-e001.gif" class="img-responsive center-block"/></div>
<p class="art-para">
In the previous equations, x and y are the distances 
measured along and perpendicular to the streamwise 
direction and u and v are the velocity components in the 
corresponding directions. C<sub>p</sub> is the specific heat; k the thermal 
conductivity; ρ the density; μ the viscosity; σ the electrical 
conductivity and B<sub>0</sub> is the magnetic field strength.</p>
<p class="art-para">Proceeding with the analysis, we define the following 
transformations:</p>
<div class="art-img" id="e002">
<img src="<?php echo $imgpath;?>images/ijae-112-e002.gif" class="img-responsive center-block"/></div>
<p class="art-para">

Here, c<sub>m</sub> and c<sub>k</sub> are chosen the same as Liu and Cao (2017).
(12)</p>
<div class="art-img" id="e003">
<img src="<?php echo $imgpath;?>images/ijae-112-e003.gif" class="img-responsive center-block"/></div>
<p class="art-para">When the temperature range is from 600 K to 3500 K, the 
molecular vibration degrees of freedom of Oxygen and 
Nitrogen molecules in the air are excited and the specific 
enthalpy can be expressed following Jia and Cao (2010) as:</p>
<div class="art-img" id="e004">
<img src="<?php echo $imgpath;?>images/ijae-112-e004.gif" class="img-responsive center-block"/></div>
<p class="art-para">where R represents the gas constant and Tve denotes the 
vibration eigen temperature.</p>
<p class="art-para"> 
At high temperatures, the Oxygen and Nitrogen molecules 
are excited to the vibrational mode. We assume that 
Tve = 3030 K following Jia and Cao (2010)</p>
<p class="art-para"> 
The specific heat C<sub>p</sub> may be written as</p>
<div class="art-img" id="e005">
<img src="<?php echo $imgpath;?>images/ijae-112-e005.gif" class="img-responsive center-block"/></div>
<p class="art-para"> 

The transformed boundary conditions may be written as
(isothermal wall); g′(0)=0 (adiabatic wall)</p>
<div class="art-img" id="e006">
<img src="<?php echo $imgpath;?>images/ijae-112-e006.gif" class="img-responsive center-block"/></div>
<p class="art-para"> 

The local wall shear stress is given by</p>
<div class="art-img" id="e007">
<img src="<?php echo $imgpath;?>images/ijae-112-e007.gif" class="img-responsive center-block"/></div>
<p class="art-para"> We define</p>
<p class="art-para"> The local friction factor may be written as</p>
<div class="art-img" id="e008">
<img src="<?php echo $imgpath;?>images/ijae-112-e008.gif" class="img-responsive center-block"/></div>
<p class="art-para"> For the isothermal wall case, the local heat flux can be written as</p>
<div class="art-img" id="e009">
<img src="<?php echo $imgpath;?>images/ijae-112-e009.gif" class="img-responsive center-block"/></div>
<p class="art-para"> In the previous equation, qw &lt; 0 indicates heat transferred to 
the surface.The local heat transfer coefficient is given by</p>
<div class="art-img" id="e010">
<img src="<?php echo $imgpath;?>images/ijae-112-e010.gif" class="img-responsive center-block"/></div>
<p class="art-para"> The local Nusselt number becomes</p>
<div class="art-img" id="e011">
<img src="<?php echo $imgpath;?>images/ijae-112-e011.gif" class="img-responsive center-block"/></div>
where
<div class="art-img" id="e012">
<img src="<?php echo $imgpath;?>images/ijae-112-e012.gif" class="img-responsive center-block"/></div>
<p class="art-subhead"> Results and Discussion</p>
<p class="art-para"> 
Equations (6) and (7) are solved numerically using the implicit 
finite difference method, Keller box method (Keller and Cebeci 
1971).</p>
<p class="art-para"><b>
Isothermal Wall Case:</b></p>
<p class="art-para"> 
Tables 1-5 show the numerical results for f ′′ (0) and 
g′ (0) with M, Ma, f<sub>w</sub>, f<sub>w</sub><sup>′</sup> and g<sub>w</sub> are prescribable 
parameters.</p>
<p class="art-para"> 
Table 1. shows that as the magnetic parameter M
increases, the wall shear stress decreases and the surface heat 
transfer rate increases. This suggests that the magnetic force 
can be used to cool the surface at high Mach number 
applications.</p>
<p class="art-para"> 
<div class="art-img" id="t001">
<img src="<?php echo $imgpath;?>images/ijae-112-t001.gif" class="img-responsive center-block"/></div>
<div class="art-img" id="t002">
<img src="<?php echo $imgpath;?>images/ijae-112-t002.gif" class="img-responsive center-block"/></div>
<p class="art-para"> Table 2. shows that as the Mach number Ma increases, 
the wall shear stress and the surface heat transfer rate 
decrease. Values of Ma above 5, corresponding to hypersonic 
speeds, produce wall heating instead of wall cooling. This is 
due to the intense viscous dissipation at high Mach numbers. 
Table 3 indicates that wall blowing ( f<sub>w</sub> &gt; 0) results in increased 
wall shear stress and surface heat transfer rate. Wall suction 
( f<sub>w</sub> &lt; 0) produces reduced wall shear stress and heat transfer 
rate. Table 4 shows that the downstream vectored surface 
mass transfer reduces wall shear stress and heat transfer rates.</p>
<p class="art-para"> 

Table 5 shows the effect of the value of surface temperature 
gw on wall shear stress and heat transfer rate. As gw increases, 
the wall shear stress decreases and heat transfer rate increases.</p>
<p class="art-para"> 
Figures 1 and 2 shows the velocity and temperature 
distributions within the boundary layer for varying values of 
the magnetic field strength.</p>
<div class="art-img" id="t003">
<img src="<?php echo $imgpath;?>images/ijae-112-t003.gif" class="img-responsive center-block"/></div>
<div class="art-img" id="t004">
<img src="<?php echo $imgpath;?>images/ijae-112-t004.gif" class="img-responsive center-block"/></div>
<div class="art-img" id="t005">
<img src="<?php echo $imgpath;?>images/ijae-112-t005.gif" class="img-responsive center-block"/></div>
<div class="art-img" id="f001">
<img src="<?php echo $imgpath;?>images/ijae-112-f001.gif" class="img-responsive center-block"/></div>
<div class="art-img" id="f002">
<img src="<?php echo $imgpath;?>images/ijae-112-f002.gif" class="img-responsive center-block"/></div>
<p class="art-para"> As the value of M increases, both velocity and temperature 
decrease within the boundary layer. Figures 3 and 4 show the 
effect of Mach number Ma on the velocity and temperature 
distributions within the boundary layer. As the Mach number 
exceeds a value of 5, the boundary layer thickness becomes 
very small and the flow “wraps around” close to the surface. 
Figures 5 and 6 show that as f<sub>w</sub> increases, the boundary layer 
thickness tends to reduce. The velocity at a distance normal to 
the surface increases whereas the temperature reduces. 
Figures 7 and 8 show that as f<sub>w</sub><sup>′</sup> increases, the boundary layer 
thickness tends to reduce. The velocity at a distance normal to 
the surface increases whereas the temperature reduces.</p>
<div class="art-img" id="f003">
<img src="<?php echo $imgpath;?>images/ijae-112-f003.gif" class="img-responsive center-block"/></div>
<div class="art-img" id="f004">
<img src="<?php echo $imgpath;?>images/ijae-112-f004.gif" class="img-responsive center-block"/></div>
<div class="art-img" id="f005">
<img src="<?php echo $imgpath;?>images/ijae-112-f005.gif" class="img-responsive center-block"/></div>
<div class="art-img" id="f006">
<img src="<?php echo $imgpath;?>images/ijae-112-f006.gif" class="img-responsive center-block"/></div>
<div class="art-img" id="f007">
<img src="<?php echo $imgpath;?>images/ijae-112-f007.gif" class="img-responsive center-block"/></div>
<div class="art-img" id="f008">
<img src="<?php echo $imgpath;?>images/ijae-112-f008.gif" class="img-responsive center-block"/></div>
<div class="art-img" id="f009">
<img src="<?php echo $imgpath;?>images/ijae-112-f009.gif" class="img-responsive center-block"/></div>
<div class="art-img" id="f010">
<img src="<?php echo $imgpath;?>images/ijae-112-f010.gif" class="img-responsive center-block"/></div>
<p class="art-para">Figures 9 and 10 display result for the effect of wall 
temperature gw on the velocity and temperature profiles. As 
gw increases, the boundary layer thickness reduces. The 
velocity at a distance normal to the surface decreases whereas 
the temperature increases.</p>
<p class="art-para"><b>
Adiabatic Wall Case:</b></p>
<p class="art-para">Tables 6-9 show the numerical results for f ′′ (0) and g(0) 
with M, M, f<sub>w</sub> and f<sub>w</sub><sup>′</sup> are prescribable parameters.</p>
<p class="art-para"> 
Table 6. shows that as the magnetic parameter M
increases, the wall shear stress increases and the surface 
temperature decreases.</p>
<div class="art-img" id="t006">
<img src="<?php echo $imgpath;?>images/ijae-112-t006.gif" class="img-responsive center-block"/></div>
<div class="art-img" id="t007">
<img src="<?php echo $imgpath;?>images/ijae-112-t007.gif" class="img-responsive center-block"/></div>

<p class="art-para">Table 7. shows that as the Mach number Ma increases, 
the wall shear stress decreases and the surface temperature 
increases. This is due to the intense viscous dissipation at high 
Mach numbers.</p>
<div class="art-img" id="t008">
<img src="<?php echo $imgpath;?>images/ijae-112-t008.gif" class="img-responsive center-block"/></div>
<p class="art-para"> 
Table 8 indicates that wall blowing ( f<sub>w</sub> &gt; 0) results in 
increased wall shear stress and surface temperature. Wall 
suction ( f<sub>w</sub> &lt; 0) produces reduced wall shear stress and 
increased surface temperature. Table 9 shows that the 
downstream vectored surface mass transfer reduces wall 
shear stress and surface temperature.</p>
<div class="art-img" id="t009">
<img src="<?php echo $imgpath;?>images/ijae-112-t009.gif" class="img-responsive center-block"/></div>
<p class="art-para"> 
Figures 11 and 12 show the velocity and temperature 
distributions within the boundary layer for varying values of 
the magnetic field strength.</p>
<div class="art-img" id="f011">
<img src="<?php echo $imgpath;?>images/ijae-112-f011.gif" class="img-responsive center-block"/></div>
<div class="art-img" id="f012">
<img src="<?php echo $imgpath;?>images/ijae-112-f012.gif" class="img-responsive center-block"/></div>
<div class="art-img" id="f013">
<img src="<?php echo $imgpath;?>images/ijae-112-f013.gif" class="img-responsive center-block"/></div>
<div class="art-img" id="f014">
<img src="<?php echo $imgpath;?>images/ijae-112-f014.gif" class="img-responsive center-block"/></div>
<p class="art-para">As the value of M increases, both velocity and temperature 
increase within the boundary layer. Figures 13 and 14 show 
the effect of Mach number Ma on the velocity and temperature 
distributions within the boundary layer. As the Mach number 
exceeds a value of 5, the boundary layer thickness becomes 
very small and the flow “wraps around” close to the surface. 
Figures 15 and 16 show that as f<sub>w</sub> increases, the boundary 
layer thickness tends to reduce. The velocity at a distance 
normal to the surface increases whereas the temperature 
reduces. Figures 17 and 18 show that as f<sub>w</sub><sup>′</sup> increases, the 
boundary layer thickness tends to reduce. The velocity in the 
boundary layer increases and surface temperature decreases 
as f<sub>w</sub><sup>′</sup> increases.</p>
<div class="art-img" id="f015">
<img src="<?php echo $imgpath;?>images/ijae-112-f015.gif" class="img-responsive center-block"/></div>
<div class="art-img" id="f016">
<img src="<?php echo $imgpath;?>images/ijae-112-f016.gif" class="img-responsive center-block"/></div>
<div class="art-img" id="f017">
<img src="<?php echo $imgpath;?>images/ijae-112-f017.gif" class="img-responsive center-block"/></div>
<div class="art-img" id="f018">
<img src="<?php echo $imgpath;?>images/ijae-112-f018.gif" class="img-responsive center-block"/></div>
<p class="art-subhead">
Concluding Remarks</p>
<p class="art-para">In this paper, a boundary layer analysis is presented for 
laminar, hypersonic flow at a stagnation point. Consideration 
is given to variable properties of air. It is shown that surface 
drag and heat transfer rates may be controlled by applying 
magnetic field and vectored surface mass transfer. The range 
of Mach numbers considered is 2.5 to 10. As the magnetic 
field strength M increases, friction factor decreases and heat 
transfer rate increases. Friction factor and surface temperature 
in the case of adiabatic surface can be reduced by applying 
vectored surface mass transfer. The present proposed 
methodology of using magnetic forces and vectored surface 
mass transfer has the potential of reduction of friction factor 
and heat transfer and hence delay the transition to turbulence.</p>
<p class="art-subhead">Acknowledgement</p>
<p class="art-para">One of the authors (MK) is thankful to the Chairperson, 
Supercomputer Education and Research Centre, Indian 
Institute of Science, Bangalore, India, for providing computational 
facility. The authors are grateful to the reviewers for their 
useful comments.</p>
<p class="art-subhead">Notation List</p>
<p class="art-para">C<sub>fx</sub> local skin friction coefficient</p>
<p class="art-para">C<sub>p</sub> specific heat</p>
<p class="art-para">E<sub>c</sub> Eckert number</p>
<p class="art-para">f nondimensional stream function</p>
<p class="art-para">g nondimensional temperature</p>
<p class="art-para">h heat transfer coefficient</p>
<p class="art-para">k thermal conductivity</p>
<p class="art-para">M Magnetic Parameter (Hartmann number)</p>
<p class="art-para">Ma Mach number</p>
<p class="art-para">Nu<sub>x</sub> Nusselt number</p>
<p class="art-para">Pr Prandtl number</p>
<p class="art-para">q heat flux</p>
<p class="art-para">r recovery factor</p>
<p class="art-para">R<sub>ex</sub> Reynolds number</p>
<p class="art-para">T temperature</p>
<p class="art-para">T<sub>ve</sub> Vibrational eigen temperature</p>
<p class="art-para">u velocity component in streamwise direction</p>
<p class="art-para">v velocity component in normal direction</p>
<p class="art-para">x coordinate along streamwise direction</p>
<p class="art-para">y coordinate normal to the surface</p>
<p class="art-para">g ratio of specific heats</p>
<p class="art-para">η nondimensional coordinate</p>
<p class="art-para">m viscosity</p>
<p class="art-para">ρ density</p>
<p class="art-para">Subscripts</p>
<p class="art-para">e conditions at the boundary layer edge</p>
<p class="art-para">w conditions at the wall</p>
<p class="art-para">∞ reference free stream conditions</p>
<p class="art-subhead" id="references">References</p>
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</div>
</div>
</div>
</section>
</div>
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