How to delete an image from the Storage laravel 5.2 folder?

I'm using this code to upload image is all working my doubt and in the condition if to delete the old image I have to leave a default.jpg image as default plus my condition if deleting this image. my question and how can I leave this image always default and continue using the deletion condition can anyone help?

 public function update_avatar(Request $request){

    if($request->hasFile('avatar')){
        $user            = Auth::user();
        $old_avatar      = $user->avatar;
        $file            = $request->file('avatar');
        $filename        = time() . '.' . $file->getClientOriginalExtension();
        $old_file_avatar = $old_avatar  . '.' . $file->getClientOriginalExtension();
        $image           = Image::make($file);

        $image->fit(250, 250, function ($constraint) {
            $constraint->aspectRatio();
        });

        Storage::put($filename, (string) $image->encode());

        $user->avatar = $filename;
        $user->save();

    }

    //delete image
    if (Auth::user() !== $filename) {
        Storage::delete($old_avatar);
    }

    return redirect('profile');
}

public function getUserImage()
{
    $user        = Auth::user();
    $old_avatar  = $user->avatar;
    $file        = Storage::disk('local')->get($old_avatar);
    return Response::make($file,200,[ 'Content-Type' => $old_avatar]);


}

I want to delete the old image and put the new one if this already makes it the issue and I do not want it to overwrite the Default.jpg image that is the default in the database for all users.

 //delete image
if (Auth::user() !== $filename) {
    Storage::delete($old_avatar);
}